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Calculus Cameron Geisler

Section 2.2 Basic Derivative Rules

Calculating derivatives using the limit definition is often tedious and sometimes difficult, especially for more complicated functions. It turns out, there are various patterns and shortcut rules for computing derivatives of various functions, i.e. differentiation rules,
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  1. Rules to differentiate specific types of functions, that you have learned about in pre-calculus.
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  2. Also, rules to differentiate various types of combinations of those functions.
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\begin{equation*} \begin{cases} \text{polynomials / powers} \quad x^n \\ \text{rational functions} \quad \frac{p(x)}{q(x)} \\ \text{trigonometric functions} \quad \sin{x}, \cos{x}, \tan{x}, \dots \\ \text{exponential functions} \quad b^x, e^x \\ \text{logarithmic functions} \quad \log{x} \\ \qquad \vdots \end{cases} \qquad \begin{cases} \text{sum and difference (adding and subtracting)} \\ \text{product (multiplying)} \\ \text{quotient (dividing)} \\ \text{composition (one function inside another)} \end{cases} \end{equation*}
This could be called systematic differentiation. Combining all of these rules will allow us to compute derivatives of every type of function that is covered in pre-calculus.
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Subsection 2.2.1 Derivative of Constant and a Line

Perhaps the most basic function is a constant function.
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Example 2.2.1. Constant Function.

Consider the function \(f(x) = 3\text{.}\) This is a constant function, because it always outputs the same value, no matter what the input is. In particular, it always outputs the value 3. Graph of \(f(x)=3\). Its graph is a horizontal line at \(y=3\text{,}\) which has slope 0, so we would expect that its derivative would be always 0. Indeed,
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\begin{equation*} \frac{d}{dx} (3) = 0 \end{equation*}
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In general, for a constant function, of the form \(f(x)=k\) (where \(k\) is some number), its derivative is 0, because its graph is a horizontal line, which has slope 0.
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Proof.

Using the definition of the derivative, we have,
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\begin{align*} f'(x) \amp= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\\ \amp= \lim_{h \to 0} \frac{k - k}{h} \amp\amp \text{as $f(x + h) = k$ and $f(x) = k$}\\ \amp= \lim_{h \to 0} \frac{0}{h}\\ \amp= \lim_{h \to 0} 0\\ \amp= 0 \end{align*}
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In short,
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\begin{equation*} \boxed{\text{"The derivative of a constant is zero."}} \end{equation*}
A slightly more complicated situation is the case of a line.
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Example 2.2.3. Linear Function.

Consider the function \(f(x) = 2x + 3\text{.}\) This is a linear function, because its graph is a line. Graph of \(f(x)=2x+3\). This line has a slope of 2, so we would expect that its derivative would be always 2.
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In general, a line has a constant slope, so its derivative is the slope of the line.
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\begin{equation*} \boxed{\text{"The derivative of a line is its slope."}} \end{equation*}

Exercise Group 2.2.1. Derivatives of Constants and Lines.

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Subsection 2.2.2 Derivative of Power Functions (The Power Rule)

The next basic type of function we will consider are power functions, where the variable is raised to a constant number.
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Example 2.2.4. Discovering the Power Rule Pattern.

You may recall from a previous section the following derivatives of \(x\text{,}\) \(x^2\text{,}\) \(x^3\text{,}\) and \(x^4\text{.}\) If not, these can be derived from the limit definition of the derivative. They are,
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\begin{align*} \frac{d}{dx} x \amp= 1\\ \frac{d}{dx} x^2 \amp= 2x\\ \frac{d}{dx} x^3 \amp= 3x^2\\ \frac{d}{dx} x^4 \amp= 4x^3 \end{align*}
Observe the pattern for the derivative of these powers of \(x\text{.}\) The derivative has:
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  • A coefficient in the front, which is the exponent of \(x\text{.}\)
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  • The variable \(x\) raised to an exponent that is one less than what it was before.
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This applies to even the first example \(\frac{d}{dx} x = 1\text{,}\) as \(x\) has an exponent of 1, so the pattern says the derivative should be \(1 \cdot x^0 = 1\) (recall: \(x^0 = 1\) for any \(x\)).
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It turns out that this rule holds in general, for any whole number (like 1, 2, 3, etc.), but in fact, for any number.
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In other words, to differentiate a function that is some power of the variable \(x\text{,}\)
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  • Bring down the exponent into the “front” and multiply it.
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  • Subtract 1 from the exponent.
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Exercise Group 2.2.2. Basic Power Rule Practice.

Remark 2.2.6.

For the reciprocal function \(f(x) = \frac{1}{x}\text{,}\) you may recall its derivative is \(f'(x) = -\frac{1}{x^2}\text{.}\) This also follows the power rule, because \(\frac{1}{x} = x^{-1}\text{.}\)
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\begin{align*} \frac{d}{dx} \frac{1}{x} \amp= \frac{d}{dx} x^{-1}\\ \amp= -x^{-2} \amp\amp \text{by the power rule}\\ \amp= -\frac{1}{x^2} \end{align*}
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Exercise Group 2.2.3. Negative Exponent Practice.

Differentiate each function.
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(a)
\(f(x) = x^{-3}\)
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Answer.
\(f'(x) = -3x^{-4} = -\frac{3}{x^4}\)
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(b)
\(f(x) = \frac{2}{x^6}\)
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Answer.
\(f'(x) = -12x^{-7} = -\frac{12}{x^7}\)
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(c)
\(f(x) = \frac{5}{x^4}\)
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Answer.
\(f'(x) = -\frac{20}{x^5}\)
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(d)
\(f(x) = cx^{-7}\text{,}\) where \(c\) is a constant
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Answer.
\(f'(x) = -7cx^{-8}\)
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Subsection 2.2.3 Derivative of a Constant Multiple

Next, it turns out that if we have a constant multiplied by a function, then that number can be kept in the front, and we can just differentiate the function as normal.
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Exercise Group 2.2.4. Constant Multiple Practice.

Differentiate.
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(f)
\(f(x) = 6\sqrt{x}\)
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Answer.
\(f'(x) = \frac{3}{\sqrt{x}}\)
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This is formalized by the constant multiple rule.
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Subsection 2.2.4 Derivative of Sums and Differences

Example 2.2.8. Differentiating Term by Term.

Consider the derivative of \(f(x) = 3x^2 + 5x - 7\text{.}\) This function has 3 terms (\(3x^2\text{,}\) \(5x\text{,}\) and \(-7\)). It turns out, if a function has multiple terms (multiple things added or subtracted together), we can find the derivative of each term separately. The derivative of \(3x^2\) is \(6x\text{,}\) the derivative of \(5x\) is 5, and the derivative of \(-7\) is 0. So, the derivative of \(f(x)\) is \(f'(x) = 6x + 5 + 0 = 6x + 5\text{.}\)
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In general, to differentiate a function with multiple terms,
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  • Differentiate each term individually (term-by-term).
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  • For any coefficients (numbers multiplied in the front), keep them in the front (they “come along for the ride”).
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These two intuitive rules are formalized by the sum rule and difference rule.
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The sum or difference rule also applies to any number of functions, not just 2 functions. In other words,
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\begin{equation*} \frac{d}{dx} \brac{f + g + h} = \frac{d}{dx} f + \frac{d}{dx} g + \frac{d}{dx} h \end{equation*}
This basically means that derivatives can be determined term-by-term.
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Subsection 2.2.5 Derivative of Polynomial Functions

We can now differentiate any polynomial function, using the previous rules.
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Exercise Group 2.2.5. Differentiating Polynomials.

Differentiate each function.
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(b)
\(f(x) = 3x^3 - 5x^2 + 7x - 2\)
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Answer.
\(f'(x) = 9x^2 - 10x + 7\)
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(c)
\(f(x) = 3x - \frac{1}{4} x^3\)
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Answer.
\(f'(x) = 3 - \frac{3}{4} x^2\)
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Subsection 2.2.6 Derivatives with Rational Exponents and Radical Functions

The power rule can also be used to differentiate functions with rational exponents (i.e. exponents which are fractions), and in general any term that can be represented as a power of the variable \(x\text{.}\)
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Remark 2.2.10.

Recall that,
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  • \(\sqrt{x} = x^{1/2}\text{,}\) \(\sqrt[3]{x} = x^{1/3}\text{,}\) and in general, \(\sqrt[n]{x} = x^{1/n}\text{.}\)
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  • \(\sqrt[n]{x^m} = x^{m/n}\text{.}\)
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Example 2.2.11. Differentiating the Square Root.

For the square root function \(f(x) = \sqrt{x}\text{,}\) you may recall its derivative is \(f'(x) = \frac{1}{2\sqrt{x}}\text{.}\) This also follows the power rule, because \(\sqrt{x} = x^{1/2}\text{,}\)
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\begin{align*} \frac{d}{dx} \sqrt{x} \amp= \frac{d}{dx} x^{1/2}\\ \amp= \frac{1}{2} x^{-1/2} \amp\amp \text{by the power rule}\\ \amp= \frac{1}{2\sqrt{x}} \end{align*}
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In general, if your derivative has negative exponents, it is good practice to rewrite the final answer with positive exponents.
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Exercise Group 2.2.6. Rational Exponents and Radicals.

Differentiate each function.
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(a)
\(f(x) = 5\sqrt{x} + \frac{1}{\sqrt{x}}\)
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Answer.
\(f'(x) = \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}}\)
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(b)
\(f(x) = 3x^2 - 5\sqrt{x}\)
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Answer.
\(f'(x) = 6x - \frac{5}{2\sqrt{x}}\)
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(c)
\(f(x) = \sqrt[3]{x}\)
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Answer.
\(f'(x) = \frac{1}{3x^{2/3}}\)
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(d)
\(f(x) = 5 + 7x^{1/2} - 2x^{-1/3}\)
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Answer.
\(f'(x) = \frac{7}{2x^{1/2}} + \frac{2}{3x^{4/3}}\)
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(e)
\(f(x) = x^{7/2}\)
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Answer.
\(f'(x) = \frac{7}{2} x^{5/2}\)
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(f)
\(f(x) = \sqrt{x^3}\)
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Answer.
\(f'(x) = \frac{3}{2} x^{1/2}\)
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(g)
\(f(x) = x^{\sqrt{2}}\)
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Hint.
The same power rule applies: bring down the exponent, and subtract 1.
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Answer.
\(f'(x) = \sqrt{2}\, x^{\sqrt{2} - 1}\)
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(h)
\(f(x) = \sqrt{x^{2+\pi}}\)
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Answer.
\(f'(x) = \frac{2+\pi}{2} x^{\pi/2}\)
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(j)
\(f(x) = \frac{\sqrt{x}}{4}\)
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Answer.
\(f'(x) = \frac{1}{8\sqrt{x}}\)
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(k)
\(f(x) = 2x^{-3/4}\)
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Answer.
\(f'(x) = -\frac{3}{2}x^{-7/4}\)
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(l)
\(f(x) = \frac{1}{\sqrt{x}} + \sqrt{x}\)
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Answer.
\(f'(x) = -\frac{1}{2x^{3/2}} + \frac{1}{2\sqrt{x}}\)
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(m)
\(f(x) = \sqrt{x} + 6\sqrt[3]{x}\)
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Answer.
\(f'(x) = \frac{1}{2\sqrt{x}} + \frac{2}{x^{2/3}}\)
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Subsection 2.2.7 Practice: Differentiate

Exercise Group 2.2.7. Positive Integer Exponents.

Differentiate each function.
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(a)
\(f(x) = 3x^5 - 6x^4 + 2\)
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Answer.
\(f'(x) = 15x^4 - 24x^3\)
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(b)
\(f(x) = x^{10} + 25x^5 - 50\)
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Answer.
\(f'(x) = 10x^9 + 125x^4\)
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(c)
\(f(x) = 2x^3 + 5x^2 - 4x - 3.75\)
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Answer.
\(f'(x) = 6x^2 + 10x - 4\)
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(d)
\(f(x) = \frac{1}{5}x^5 + \frac{1}{3}x^3 - \frac{1}{2}x^2 + 1\)
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Answer.
\(f'(x) = x^4 + x^2 - x\)
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(e)
\(f(x) = \frac{26}{5}x + \frac{23}{10}\)
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Answer.
\(f'(x) = \frac{26}{5}\)
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(f)
\(f(x) = \frac{7}{4}x^2 - 3x + 12\)
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Answer.
\(f'(x) = \frac{7}{2}x - 3\)
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(g)
\(f(x) = 2x^3 - 3x^2 - 4x\)
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Answer.
\(f'(x) = 6x^2 - 6x - 4\)
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(h)
\(f(x) = \frac{7}{5}x^5 - \frac{5}{2}x^2 + \frac{67}{10}\)
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Answer.
\(f'(x) = 7x^4 - 5x\)
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(k)
\(f(x) = 6x^5 - \frac{5}{2}x^2 + x + 5\)
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Answer.
\(f'(x) = 30x^4 - 5x + 1\)
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(l)
\(f(x) = 2x^3 + 3x^2 + 10x\)
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Answer.
\(f'(x) = 6x^2 + 6x + 10\)
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(m)
\(f(x) = 6x^4 - 5x^3 - 2x + 17\)
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Answer.
\(f'(x) = 24x^3 - 15x^2 - 2\)
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Exercise Group 2.2.8. Negative Exponents.

Differentiate each function.
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(a)
\(f(x) = x^2 - \frac{2}{x^2}\)
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Answer.
\(f'(x) = 2x + \frac{4}{x^3}\)
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(b)
\(y = x^5 - 6x^{-5}\)
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Answer.
\(\frac{dy}{dx} = 5x^4 + \frac{30}{x^6}\)
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(c)
\(y = \frac{6}{x^3} + \frac{2}{x^2} - 3\)
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Answer.
\(y' = -\frac{18}{x^4} - \frac{4}{x^3}\)
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Exercise Group 2.2.9. Fractional Exponents and Radicals.

Differentiate each function.
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(a)
\(f(x) = \sqrt{x} - 5x^4\)
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Answer.
\(f'(x) = \frac{1}{2\sqrt{x}} - 20x^3\)
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(b)
\(f(x) = \sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}\)
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Answer.
\(f'(x) = \frac{1}{2\sqrt{x}} + \frac{1}{3x^{2/3}} + \frac{1}{4x^{3/4}}\)
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(c)
\(f(x) = x^{5/3} - x^{2/3}\)
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Answer.
\(f'(x) = \frac{5}{3}x^{2/3} - \frac{2}{3}x^{-1/3}\)
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(d)
\(f(x) = \sqrt{x} - x\)
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Answer.
\(f'(x) = \frac{1}{2\sqrt{x}} - 1\)
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(e)
\(y = 4x^{-1/2} - \frac{6}{x}\)
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Answer.
\(y' = -\frac{2}{x^{3/2}} + \frac{6}{x^2}\)
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(f)
\(y = 9x^{-2} + 3\sqrt{x}\)
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Answer.
\(y' = -\frac{18}{x^3} + \frac{3}{2\sqrt{x}}\)
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(g)
\(y = \sqrt{x} + 6\sqrt{x^3} + \sqrt{2}\)
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Answer.
\(y' = \frac{1}{2\sqrt{x}} + 9\sqrt{x}\)
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(h)
\(f(x) = 6\sqrt{x} - 4x^3 + 9\)
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Answer.
\(f'(x) = \frac{3}{\sqrt{x}} - 12x^2\)
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(i)
\(f(x) = 4\sqrt{x} - \frac{1}{4}x^4 + x + 1\)
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Answer.
\(f'(x) = \frac{2}{\sqrt{x}} - x^3 + 1\)
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(j)
\(f(x) = 2x^4 + \sqrt{x}\)
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Answer.
\(f'(x) = 8x^3 + \frac{1}{2\sqrt{x}}\)
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Exercise Group 2.2.10. Evaluating the Derivative.

Determine \(f'(a)\) for the given function \(f(x)\) at the given value of \(a\text{.}\)
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(a)
\(f(x) = x^3 - \sqrt{x}\text{,}\) \(a = 4\)
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Answer.
\(f'(4) = \frac{191}{4}\)
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(b)
\(f(x) = 7 - 6\sqrt{x} + 5x^{2/3}\text{,}\) \(a = 64\)
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Answer.
\(f'(64) = \frac{11}{24}\)
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Exercise Group 2.2.11. Slope of the Tangent.

Determine the slope of the tangent to each curve at the given point.
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(c)
\(y = \frac{2}{x}\text{,}\) \((-2, -1)\)
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Answer.
\(-\frac{1}{2}\)
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(f)
\(y = 2\sqrt{x} + 5\text{,}\) \(x = 4\)
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Answer.
\(\frac{1}{2}\)
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(h)
\(y = x^{-3}\brac{x^{-1} + 1}\text{,}\) \(x = 1\)
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Answer.
\(-7\)
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Subsection 2.2.8 Practice: Tangent Lines

Exercise Group 2.2.12. Tangent Line Equations.

Find an equation of the tangent line to each function at the given point.
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(a)
\(f(x) = x^2\) at \((1, 1)\)
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Answer.
\(y - 1 = 2(x-1)\) or \(y = 2x - 1\)
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(b)
\(y = x^3 + 4\) at the point \((1, 5)\)
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Answer.
\(y - 5 = 3(x-1)\) or \(y = 3x + 2\)
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(c)
\(f(x) = \sqrt{x}\) at \((4, 2)\)
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Answer.
\(y - 2 = \frac{1}{4}(x-4)\) or \(y = \frac{1}{4}x + 1\)
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(d)
\(f(x) = 5\sqrt[3]{x}\) at \(x = 64\)
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Answer.
\(y - 20 = \frac{5}{48}(x-64)\) or \(y = \frac{5}{48}x + \frac{40}{3}\)
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(e)
\(y = x^3 - x^2 + x - 1\text{,}\) \((1, 0)\)
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Answer.
\(y = 2x - 2\)
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(f)
\(y = 7\sqrt{x} - 3x\text{,}\) \((1, 4)\)
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Answer.
\(y = \frac{1}{2}x + \frac{7}{2}\)
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(g)
\(y = x + \frac{6}{x}\text{,}\) \((2, 5)\)
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Answer.
\(y = -\frac{1}{2}x + 6\)
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(h)
\(y = (x^2+1)^2\text{,}\) \((-1, 4)\)
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Answer.
\(y = -8x - 4\)
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(i)
\(y = 2x - \frac{1}{x}\) at \(P(0.5, -1)\)
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Answer.
\(y = 6x - 4\)
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(j)
\(y = \frac{3}{x^2} - \frac{4}{x^3}\) at \(P(-1, 7)\)
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Answer.
\(y = 18x + 25\)
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(k)
\(y = \sqrt{3x^3}\) at \(P(3, 9)\)
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Answer.
\(y = \frac{9}{2}x - \frac{9}{2}\)
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(l)
\(y = \frac{1}{x}\brac{x^2 + \frac{1}{x}}\) at \(P(1, 2)\)
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Answer.
\(y = -x + 3\)
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(m)
\(y = \brac{\sqrt{x} - 2}\brac{3\sqrt{x} + 8}\) at \(P(4, 0)\)
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Answer.
\(y = \frac{7}{2}x - 14\)
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(n)
\(y = \frac{\sqrt{x} - 2}{\sqrt[3]{x}}\) at \(P(1, -1)\)
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Answer.
\(y = \frac{5}{6}x - \frac{11}{6}\)
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Exercise Group 2.2.13. Horizontal Tangent Lines.

Find all points on the graph where the tangent line is horizontal.
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(a)
\(f(x) = -x^3 + 3x^2 - 2\)
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Answer.
\((0, -2)\) and \((2, 2)\)
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(b)
\(f(x) = x^4 - 2x^2 + 2\)
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Answer.
\((-1, 1)\text{,}\) \((0, 2)\text{,}\) \((1, 1)\)
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(c)
\(y = x^3 + 3x^2 - 24x + 1\)
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Answer.
\((-4, 81)\) and \((2, -27)\)
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Subsection 2.2.9 Simplifying Before Using Basic Rules

Sometimes, you need to rewrite the function first before you can use the basic rules. In particular, write it as a sum of multiple terms.
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Exercise Group 2.2.14. Expand Then Differentiate.

Differentiate each function.
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(b)
\(f(x) = x(x^2 + 3x - 9)\)
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Answer.
\(f'(x) = 3x^2 + 6x - 9\)
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(c)
\(f(x) = \brac{\frac{x}{2}}^4\)
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Answer.
\(f'(x) = \frac{x^3}{4}\)
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(e)
\(f(x) = (2x+1)(3x^2+2)\)
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Answer.
\(f'(x) = 18x^2 + 6x + 4\)
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Exercise Group 2.2.15. Divide Then Differentiate.

Differentiate each function.
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(a)
\(f(x) = \frac{x^2-6}{2x}\)
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Answer.
\(f'(x) = \frac{1}{2} + \frac{3}{x^2}\)
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(c)
\(f(x) = \frac{6x^2-5x}{x^3}\)
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Answer.
\(f'(x) = -\frac{6}{x^2} + \frac{10}{x^3}\)
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(d)
\(f(x) = \sqrt{x}\brac{\sqrt{x} - x^{3/2}}\)
🔗
Answer.
\(f'(x) = 1 - 2x\)
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🔗
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(e)
\(f(x) = \frac{x^2 + 5\sqrt{x}}{x^2}\)
🔗
Answer.
\(f'(x) = -\frac{15}{2x^{5/2}}\)
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(h)
\(f(x) = \frac{12x^3-8x^2+12x}{4x}\)
🔗
Answer.
\(f'(x) = 6x - 2\)
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🔗
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(i)
\(f(x) = \frac{x+1}{\sqrt{x}}\)
🔗
Answer.
\(f'(x) = \frac{1}{2\sqrt{x}} - \frac{1}{2x^{3/2}}\)
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🔗
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(k)
\(f(x) = \frac{x^5-3x^2}{2x}\text{,}\) \(x \gt 0\)
🔗
Answer.
\(f'(x) = 2x^3 - \frac{3}{2}\)
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(n)
\(f(x) = \frac{1+\sqrt{x}}{x}\)
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Answer.
\(f'(x) = -\frac{1}{x^2} - \frac{1}{2x^{3/2}}\)
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Exercise Group 2.2.16. Mixed Simplification.

Differentiate each function.
🔗
(a)
\(f(x) = \brac{x - \frac{2}{\sqrt{x}}}^2\)
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Answer.
\(f'(x) = 2x - \frac{2}{\sqrt{x}} - \frac{4}{x^2}\)
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🔗
🔗
(b)
\(f(x) = \sqrt{x}(2+3x)\)
🔗
Answer.
\(f'(x) = \frac{1}{\sqrt{x}} + \frac{9\sqrt{x}}{2}\)
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🔗
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(c)
\(f(x) = (5x^3+3x+1)(x^2+3)\)
🔗
Answer.
\(f'(x) = 25x^4 + 54x^2 + 2x + 9\)
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🔗
🔗
(d)
\(f(x) = \sqrt[3]{x}(2+x)\)
🔗
Answer.
\(f'(x) = \frac{4}{3}x^{1/3} + \frac{2}{3}x^{-2/3}\)
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🔗
🔗
(e)
\(f(x) = \frac{\sqrt{x}+x}{x^2}\)
🔗
Answer.
\(f'(x) = -\frac{3}{2}x^{-5/2} - x^{-2}\)
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🔗
🔗
(g)
\(f(x) = \frac{x^2+4x+3}{\sqrt{x}}\)
🔗
Answer.
\(f'(x) = \frac{3}{2}\sqrt{x} + \frac{2}{\sqrt{x}} - \frac{3}{2}x^{-3/2}\)
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🔗
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🔗

Exercise Group 2.2.17. Abstract Functions.

Differentiate each function.
🔗
(a)
\(f(x) = \sqrt{5x} + \frac{\sqrt{7}}{x}\)
🔗
Answer.
\(f'(x) = \frac{\sqrt{5}}{2\sqrt{x}} - \frac{\sqrt{7}}{x^2}\)
🔗
🔗
🔗
(b)
\(f(x) = (1+x^{-1})^2\)
🔗
Answer.
\(f'(x) = -\frac{2}{x^2} - \frac{2}{x^3}\)
🔗
🔗
🔗
(c)
\(f(x) = \frac{A+Bx+Cx^2}{x^2}\)
🔗
Answer.
\(f'(x) = -\frac{2A}{x^3} - \frac{B}{x^2}\)
🔗
🔗
🔗
(d)
\(f(x) = \frac{1+16x^2}{(4x)^3}\)
🔗
Answer.
\(f'(x) = -\frac{3}{64x^4} - \frac{1}{4x^2}\)
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🔗
🔗
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🔗

Subsection 2.2.10 Examples

Checkpoint 2.2.12. Parabola Slope.

At what point on the parabola \(y = 3x^2\) is the slope of the tangent line equal to 24?
🔗
Answer.
\((4, 48)\)
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🔗

Checkpoint 2.2.13. Parallel Tangent on Hyperbola.

Find all points on the graph of \(y = \frac{12}{x}\) where the tangent line is parallel to the line \(3x+y=0\text{.}\)
🔗
Answer.
\((2, 6)\) and \((-2, -6)\)
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🔗

Checkpoint 2.2.14. Parallel Tangent on Quartic.

Find all points on the graph of \(y = x^4 - 25x + 2\) where the tangent line is parallel to the line \(7x - y = 2\text{.}\)
🔗
Answer.
\((2, -32)\)
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Checkpoint 2.2.15. Parallel Tangent on Cube Root.

Find all points on the graph of \(y = \frac{3}{\sqrt[3]{x}}\) where the tangent line is parallel to the line \(x + 16y + 3 = 0\text{.}\)
🔗
Answer.
\(\brac{-8,-\frac{3}{2}}\) and \(\brac{8,\frac{3}{2}}\)
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🔗

Checkpoint 2.2.16. Parallel Tangent on Parabola.

Find all points on the graph of \(f(x) = 3x^2 - 4x\) where the tangent line is parallel to the line \(y = 8x + 5\text{.}\)
🔗
Answer.
\((2, 4)\)
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Checkpoint 2.2.17. Parallel Tangent on Cubic.

Find all points on the graph of \(f(x) = \frac{1}{3}x^3 - \frac{3}{2}x^2 + 1\) where the tangent line is parallel to the line \(8x - 2y = 1\text{.}\)
🔗
Answer.
\(\brac{-1, -\frac{5}{6}}\) and \(\brac{4, -\frac{5}{3}}\)
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🔗
🔗

Checkpoint 2.2.18. Parallel Tangent on Power Function.

Find the point on the curve \(y = x\sqrt{x}\) where the tangent line is parallel to the line \(6x - y = 4\text{.}\)
🔗
Answer.
\((16, 64)\)
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🔗
🔗

Checkpoint 2.2.19. No Tangent with Given Slope.

Show that the curve \(y = 10x^3 + 4x + 2\) has no tangent lines with slope 3.
🔗
Hint.
Solve \(y' = 30x^2 + 4 = 3\text{.}\)
🔗
🔗
Answer.
The equation becomes \(x^2 = -\frac{1}{30}\text{,}\) which has no real solution.
🔗
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🔗

Checkpoint 2.2.20. Normal Line.

Determine the equation of the normal to the graph of \(y = \frac{3}{x^2} - \frac{4}{x^3}\) at \((-1, 7)\text{.}\) (A normal line is perpendicular to the tangent line at the point of tangency.)
🔗
Answer.
The tangent slope is 18, so the normal slope is \(-\frac{1}{18}\) and the normal line is \(y = -\frac{1}{18}x + \frac{125}{18}\text{.}\)
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🔗
🔗

Checkpoint 2.2.21. Same Slope?

Do the functions \(y = \frac{1}{x}\) and \(y = x^3\) ever have the same slope? If so, where?
🔗
Answer.
No, they never have the same slope (there is no real solution).
🔗
🔗
🔗

Checkpoint 2.2.22. Perpendicular Tangents.

Show that the tangent lines to \(y = x^2\) at \((2, 4)\) and \(\brac{-\frac{1}{8}, \frac{1}{64}}\) are perpendicular.
🔗
Answer.
The slopes of the tangent lines are \(4\) and \(-\frac{1}{4}\text{,}\) and since \(4 \cdot \brac{-\frac{1}{4}} = -1\text{,}\) the tangent lines are perpendicular.
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🔗
🔗

Checkpoint 2.2.23. Point Where Slope Equals 5.

Determine the point on the parabola \(y = -x^2 + 3x + 4\) where the slope of the tangent is 5.
🔗
Answer.
\((-1, 0)\)
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🔗
🔗

Checkpoint 2.2.24. Points Where Slope Equals 12.

Determine the coordinates of the points on the graph of \(y = x^3 + 2\) at which the slope of the tangent is 12.
🔗
Answer.
\((-2, -6)\) and \((2, 10)\)
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🔗
🔗

Checkpoint 2.2.25. Two Tangents with Given Slope.

Show that there are two tangents to the curve \(y = \frac{1}{5}x^5 - 10x\) that have a slope of 6.
🔗
Answer.
The tangency points occur at \(x = -2\) and \(x = 2\text{,}\) giving tangent lines \(y = 6x + \frac{128}{5}\) and \(y = 6x - \frac{128}{5}\text{.}\)
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🔗
🔗

Checkpoint 2.2.26. Find the Constant.

Determine the value of \(a\text{,}\) given that the line \(ax - 4y + 21 = 0\) is tangent to the graph of \(y = \frac{a}{x^2}\) at \(x = -2\text{.}\)
🔗
Answer.
\(a = 7\)
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🔗
🔗

Checkpoint 2.2.27. Slope Locations on Parabola.

(a)
Find the values of \(x\) for which the slope of the curve \(y = f(x)\) is 0.
🔗
Answer.
\(x = 3\)
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🔗
🔗
(b)
Find the values of \(x\) for which the slope of the curve \(y = f(x)\) is 2.
🔗
Answer.
\(x = 4\)
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🔗
🔗
🔗

Checkpoint 2.2.28. Slope Locations on Cubic.

(a)
Find the values of \(x\) for which the slope of the curve \(y = f(x)\) is 0.
🔗
Answer.
\(x = -3, 3\)
🔗
🔗
🔗
(b)
Find the values of \(x\) for which the slope of the curve \(y = f(x)\) is 21.
🔗
Answer.
\(x = -4, 4\)
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🔗
🔗
🔗

Checkpoint 2.2.29. Tangent Slope Conditions.

(a)
Find all points on the graph of \(f\) at which the tangent line is horizontal.
🔗
Answer.
\((-1, 11)\) and \((2, -16)\)
🔗
🔗
🔗
(b)
Find all points on the graph of \(f\) at which the tangent line has slope 60.
🔗
Answer.
\((-3, -41)\) and \((4, 36)\)
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🔗
🔗
🔗

Checkpoint 2.2.30. Tangent Slope Conditions, Radical Function.

(a)
Find all points on the graph of \(f\) at which the tangent line is horizontal.
🔗
Answer.
\((4, 4)\)
🔗
🔗
🔗
(b)
Find all points on the graph of \(f\) at which the tangent line has slope \(-\frac{1}{2}\text{.}\)
🔗
Answer.
\((16, 0)\)
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🔗
🔗
🔗

Checkpoint 2.2.31. Find the Parabola.

Find the parabola with equation \(y = ax^2 + bx\) whose tangent line at \((1, 1)\) has equation \(y = 3x - 2\text{.}\)
🔗
Hint.
Use \(f(1) = 1\) and \(f'(1) = 3\text{.}\)
🔗
🔗
Answer.
\(y = 2x^2 - x\)
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🔗
🔗

Checkpoint 2.2.32. Find Constants for Tangent.

For what values of \(a\) and \(b\) is the line \(2x + y = b\) tangent to the parabola \(y = ax^2\) when \(x = 2\text{?}\)
🔗
Hint.
Match the slope and the point at \(x = 2\text{.}\)
🔗
🔗
Answer.
\(a = -\frac{1}{2}\) and \(b = 2\)
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🔗
🔗

Checkpoint 2.2.33. Find p and q.

The graph of \(f(x) = \frac{p}{\sqrt{x}} + q\sqrt{x}\) has a horizontal tangent line at the point \((4, 12)\text{.}\) Find the values of \(p\) and \(q\text{.}\)
🔗
Hint.
Use \(f(4) = 12\) and \(f'(4) = 0\text{.}\)
🔗
🔗
Answer.
\(p = 12\) and \(q = 3\)
🔗
🔗
🔗

Checkpoint 2.2.34. Find a and b for Tangent.

Determine the values of \(a\) and \(b\) such that the line \(y = 5x + 6\) is tangent to the graph of \(f(x) = ax - \frac{b}{x}\) at the point where \(x = 1\text{.}\)
🔗
Hint.
Use \(f(1) = 11\) and \(f'(1) = 5\text{.}\)
🔗
🔗
Answer.
\(a = 8\) and \(b = -3\)
🔗
🔗
🔗

Checkpoint 2.2.35. Tangent at Origin, Second Intersection.

(a)
Find an equation for the line that is tangent to the curve \(y = x^3 - 6x^2 + 5x\) at the origin.
🔗
Hint.
Use \(y' = 3x^2 - 12x + 5\) and evaluate at \(x = 0\text{.}\)
🔗
🔗
Answer.
\(y = 5x\)
🔗
🔗
🔗
(b)
The tangent line from part (a) intersects the curve at another point. Find the coordinates of this point.
🔗
Hint.
Solve \(x^3 - 6x^2 + 5x = 5x\text{.}\)
🔗
🔗
Answer.
\((6, 30)\)
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🔗
🔗
🔗
🔗

Subsection 2.2.11 Examples: Tangent Lines at an Arbitrary Point

Checkpoint 2.2.36. Tangent Through External Point.

Find the equation of the tangent line(s) to \(y = x^2\) that passes through the point \((1, -3)\text{.}\)
🔗
Hint.
Find the tangent line at an arbitrary point (say, \(a\)), and then find the value of \(a\) such that the tangent line passes through \((1, -3)\text{.}\)
🔗
🔗
Answer.
\(y - 9 = 6(x-3)\) and \(y - 1 = -2(x+1)\)
🔗
🔗
🔗

Checkpoint 2.2.37. Tangent Through External Point II.

Find the equation of the tangent line(s) to \(y = 3x^2\) that passes through the point \((2, 9)\text{.}\)
🔗
Answer.
\(y - 3 = 6(x-1)\) and \(y - 27 = 18(x-3)\)
🔗
🔗
🔗

Checkpoint 2.2.38. Tangent Through Point on Hyperbola.

The tangent at point \(P\) on the curve \(y = \frac{1}{x}\) passes through \((4, 0)\text{.}\) Find the coordinates of \(P\text{.}\)
🔗
Answer.
\(\brac{2, \frac{1}{2}}\)
🔗
🔗
🔗

Checkpoint 2.2.39. Tangent Lines Through External Point, Parabola.

Find the equations of both lines that pass through the point \(P(2, 9)\) and are tangent to the parabola \(y = 2x - x^2\text{.}\)
🔗
Answer.
\(y = -8x + 25\) and \(y = 4x + 1\)
🔗
🔗
🔗

Checkpoint 2.2.40. Tangent Lines Through Origin.

Find the equations of both lines that pass through the origin and are tangent to the parabola \(y = 1 + x^2\text{.}\)
🔗
Answer.
\(y = 2x\) and \(y = -2x\)
🔗
🔗
🔗

Checkpoint 2.2.41. Tangent with Given Slope.

Find the equation of the line tangent to the curve \(y = x + \sqrt{x}\) that has slope 2.
🔗
Hint.
The tangent at \(x = a\) has slope \(1 + \frac{1}{2\sqrt{a}}\text{,}\) which must equal 2.
🔗
🔗
Answer.
\(y = 2x + \frac{1}{4}\)
🔗
🔗
🔗

Checkpoint 2.2.42. Tangent Intersection Points.

Find the \(x\)-coordinates of the points on the curve \(y = \frac{1}{x}\) where the tangents from the point \((1, -1)\) intersect the curve.
🔗
Hint.
The tangent at \(x = a\) has equation \(y - \frac{1}{a} = -\frac{1}{a^2}(x - a)\text{,}\) and it must pass through \((1, -1)\text{.}\)
🔗
🔗
Answer.
\(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\)
🔗
🔗
🔗

Checkpoint 2.2.43. Tangents to Parabola Through Given Points.

Checkpoint 2.2.44. Tangent Through Point, Parabola.

Find the coordinates of the points on the parabola \(y = x^2\) where its tangent line passes through the point \((0, -4)\text{.}\)
🔗
Hint.
The tangent to \(y = x^2\) at \(x = a\) is \(y - a^2 = 2a(x-a)\text{,}\) and it must pass through \((0, -4)\text{.}\)
🔗
🔗
Answer.
The points are \((-2, 4)\) and \((2, 4)\text{.}\)
🔗
🔗
🔗

Checkpoint 2.2.45. Lines Through Point Tangent to Parabola.

(a)
Find equations of both lines through the point \((2, -3)\) that are tangent to the parabola \(y = x^2 + x\text{.}\)
🔗
Hint.
The tangent at \(x = a\) has equation \(y = (2a+1)x - a^2\text{,}\) and it must pass through \((2, -3)\text{.}\)
🔗
🔗
Answer.
\(y = -x - 1\) and \(y = 11x - 25\)
🔗
🔗
🔗
(b)
Show that there is no line through the point \((2, 7)\) that is tangent to the parabola.
🔗
Hint.
Use \(y = (2a+1)x - a^2\) and require it to pass through \((2, 7)\text{,}\) then check whether \(a\) is real.
🔗
🔗
Answer.
No tangent line exists.
🔗
🔗
🔗
🔗

Checkpoint 2.2.46. Y-Intercept of Tangent.

Let \(k\) be the \(y\)-intercept of the tangent line to the graph of \(f(x) = x^2 + x\) passing through the point \((1, -2)\text{.}\) Find the value of \(k\text{.}\)
🔗
Hint.
The tangent at \(x = a\) is \(y = (2a+1)x - a^2\text{,}\) and it must pass through \((1, -2)\text{.}\)
🔗
🔗
Answer.
\(k = -1\) or \(k = -9\)
🔗
🔗
🔗

Checkpoint 2.2.47. Points with Tangent Through Given Point.

Find all points \((x, y)\) on the graph of \(f(x) = x^2\) with tangent lines passing through the point \((3, 8)\text{.}\)
🔗
Answer.
\((2, 4)\) and \((4, 16)\)
🔗
🔗
🔗
🔗

Subsection 2.2.12 Practice: Advanced

Checkpoint 2.2.48. Steeper Tangent.

Find all values of \(x\) such that the tangent line to the graph of \(f(x) = 3x^2 - 5x + 4\) is steeper than the tangent line of \(g(x) = \frac{1}{3}x^3\text{.}\)
🔗
Hint.
\(f'(x) \gt g'(x)\) leads to a quadratic inequality \(x^2 - 6x + 5 \lt 0\text{.}\)
🔗
🔗
Answer.
\(1 \lt x \lt 5\text{,}\) or the interval \((1, 5)\)
🔗
🔗
🔗

Checkpoint 2.2.49. Y-Intercept from X-Intercept.

A tangent line to the graph of \(f(x) = 2\sqrt{x}\) has an \(x\)-intercept of \(-9\text{.}\) Find the \(y\)-intercept of this tangent line.
🔗
Hint.
The tangent at \(x = a\) has equation \(y = \frac{1}{\sqrt{a}}x + \sqrt{a}\text{,}\) so its \(x\)-intercept is \(-a\text{.}\)
🔗
🔗
Answer.
The \(y\)-intercept is 3.
🔗
🔗
🔗

Checkpoint 2.2.50. Find c: Line Tangent to Radical Curve.

Find the value of \(c\) such that the line \(y = \frac{3}{2}x + 6\) is tangent to the curve \(y = c\sqrt{x}\text{.}\)
🔗
Hint.
At the tangency point \(x = a\text{,}\) solve \(\frac{c}{2\sqrt{a}} = \frac{3}{2}\) and \(c\sqrt{a} = \frac{3}{2}a + 6\text{.}\)
🔗
🔗
Answer.
\(c = 6\)
🔗
🔗
🔗

Checkpoint 2.2.51. Find c: Line Tangent to Parabola.

What is the value of \(c\) such that the line \(y = 2x + 3\) is tangent to the parabola \(y = cx^2\text{?}\)
🔗
Hint.
At the tangency point \(x = a\text{,}\) solve \(2ca = 2\) and \(ca^2 = 2a + 3\text{.}\)
🔗
🔗
Answer.
\(c = -\frac{1}{3}\)
🔗
🔗
🔗

Checkpoint 2.2.52. Tangent at Point, Second Intersection.

(a)
Find an equation for the line that is tangent to the curve \(y = x^3 - x\) at the point \((-1, 0)\text{.}\)
🔗
Hint.
Use \(y' = 3x^2 - 1\) and \(y - y_0 = m(x - x_0)\text{.}\)
🔗
🔗
Answer.
\(y = 2x + 2\)
🔗
🔗
🔗
(b)
The tangent line from part (a) intersects the curve at another point. Find the coordinates of this point.
🔗
Hint.
Solve \(x^3 - x = 2x + 2\text{.}\)
🔗
🔗
Answer.
\((2, 6)\)
🔗
🔗
🔗
🔗

Checkpoint 2.2.53. Abstract Function Differentiation.

Differentiate each function.
🔗
(a)
\(f(x) = Ax^3 + Bx^2 + Cx\)
🔗
Answer.
\(f'(x) = 3Ax^2 + 2Bx + C\)
🔗
🔗
🔗
(b)
\(f(x) = a + \frac{b}{x} + \frac{c}{x^2}\)
🔗
Answer.
\(f'(x) = -\frac{b}{x^2} - \frac{2c}{x^3}\)
🔗
🔗
🔗
🔗

Checkpoint 2.2.54. Simplify Then Differentiate.

Differentiate each function.
🔗
(b)
\(f(x) = \frac{x^3-6x^2+8x}{x^2-2x}\)
🔗
Answer.
\(f'(x) = 1\)
🔗
🔗
🔗
(c)
\(f(x) = \frac{x-a}{\sqrt{x}-\sqrt{a}}\text{,}\) where \(a\) is a positive constant
🔗
Answer.
\(f'(x) = \frac{1}{2\sqrt{x}}\)
🔗
🔗
🔗
(d)
\(f(x) = \frac{x^2-2ax+a^2}{x-a}\text{,}\) where \(a\) is a constant
🔗
Answer.
\(f'(x) = 1\)
🔗
🔗
🔗
🔗

Checkpoint 2.2.55. Given Function Values.

Given that \(f(4) = 2\text{,}\) \(g(4) = 5\text{,}\) \(f'(4) = 6\text{,}\) and \(g'(4) = -3\text{,}\) find \((3f+8g)'(4)\text{.}\)
🔗
Hint.
\(= 3f'(4) + 8g'(4) = 3(6) + 8(-3)\text{.}\)
🔗
🔗
Answer.
\(-6\)
🔗
🔗
🔗

Checkpoint 2.2.56. Point on Curve with Given Tangent.

Find the point on the curve \(y = \frac{16}{x^2} - 1\) that has a tangent line with equation \(4x - y + 11 = 0\text{.}\)
🔗
Hint.
Compare the slope of the line with the derivative of the curve, then check that the point lies on both graphs.
🔗
🔗
Answer.
The line is \(y = 4x + 11\text{,}\) so its slope is 4. For the curve, \(y' = -\frac{32}{x^3}\text{.}\) Setting \(-\frac{32}{x^3} = 4\) gives \(x = -2\text{.}\) Then \(y = \frac{16}{(-2)^2} - 1 = 3\text{,}\) and the line also gives \(y = 4(-2) + 11 = 3\text{.}\) So the line is tangent to the curve at \((-2, 3)\text{.}\)
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Checkpoint 2.2.57. Tangent Lines Using Known Values.

Suppose \(f(3) = 1\) and \(f'(3) = 4\text{.}\) Let \(g(x) = x^2 + f(x)\) and \(h(x) = 3f(x)\text{.}\)
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(a)
Find an equation of the line tangent to \(y = g(x)\) at \(x = 3\text{.}\)
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Answer.
\(y = 10x - 20\)
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(b)
Find an equation of the line tangent to \(y = h(x)\) at \(x = 3\text{.}\)
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Answer.
\(y = 12x - 33\)
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Checkpoint 2.2.58. Tangent Lines from Known Tangent Lines.

Suppose the line tangent to the graph of \(f\) at \(x = 2\) is \(y = 4x + 1\) and the line tangent to the graph of \(g\) at \(x = 2\) has slope 3 and passes through \((0, -2)\text{.}\) Find an equation of the line tangent to the following curves at \(x = 2\text{.}\)
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Checkpoint 2.2.59. Determine Constants b and c.

Determine the constants \(b\) and \(c\) such that the line tangent to \(f(x) = x^2 + bx + c\) at \(x = 1\) is \(y = 4x + 2\text{.}\)
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Hint.
The tangent line at \(x = 1\) has slope \(2 + b\) and passes through the point \((1, 1+b+c)\text{.}\) So, we require \(2+b=4\) and \(1+b+c=6\text{.}\)
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Answer.
\(b = 2\text{,}\) \(c = 3\)
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Checkpoint 2.2.60. Find a Cubic Function.

Find a cubic function \(y = ax^3 + bx^2 + cx + d\) whose graph has horizontal tangents at the points \((-2, 6)\) and \((2, 0)\text{.}\)
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Hint.
Use \(f(-2) = 6\text{,}\) \(f(2) = 0\text{,}\) and \(f'(-2) = 0\text{,}\) \(f'(2) = 0\text{.}\)
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Answer.
\(y = \frac{3}{16}x^3 - \frac{9}{4}x + 3\)
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Checkpoint 2.2.61. Find a Parabola from Slope Conditions.

Find a parabola with equation \(y = ax^2 + bx + c\) that has slope 4 at \(x = 1\text{,}\) slope \(-8\) at \(x = -1\text{,}\) and passes through the point \((2, 15)\text{.}\)
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Hint.
Use \(y'(x) = 2ax + b\) with \(y'(1) = 4\) and \(y'(-1) = -8\text{,}\) then use \(y(2) = 15\text{.}\)
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Answer.
\(y = 3x^2 - 2x + 7\)
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Checkpoint 2.2.62. Find a,b,c,d from Tangent Conditions.

Suppose the curve \(y = x^4 + ax^3 + bx^2 + cx + d\) has a tangent line when \(x = 0\) with equation \(y = 2x + 1\) and a tangent line when \(x = 1\) with equation \(y = 2 - 3x\text{.}\) Find the values of \(a, b, c, d\text{.}\)
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Hint.
Use \(f(0) = 1\text{,}\) \(f'(0) = 2\text{,}\) \(f(1) = -1\text{,}\) and \(f'(1) = -3\text{.}\)
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Answer.
\(a = 1\text{,}\) \(b = -6\text{,}\) \(c = 2\text{,}\) \(d = 1\)
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Checkpoint 2.2.63. Tangent to Line at Origin.

The curve \(y = ax^2 + bx + c\) passes through the point \((1, 2)\) and is tangent to the line \(y = x\) at the origin. Find \(a, b, c\text{.}\)
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Answer.
\(a = 1\text{,}\) \(b = 1\text{,}\) \(c = 0\)
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Checkpoint 2.2.64. Quadratic with Horizontal Tangent.

Determine a quadratic function \(f(x) = ax^2 + bx + c\) if its graph passes through the point \((2, 19)\) and it has a horizontal tangent at \((-1, -8)\text{.}\)
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Hint.
Use \(f(2) = 19\text{,}\) \(f(-1) = -8\text{,}\) and \(f'(-1) = 0\text{.}\)
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Answer.
\(f(x) = 3x^2 + 6x - 5\)
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Checkpoint 2.2.65. Common Tangent of Two Curves.

The curves \(y = x^2 + ax + b\) and \(y = cx - x^2\) have a common tangent line at the point \((1, 0)\text{.}\) Find \(a, b, c\text{.}\)
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Hint.
Use \(f(1) = 0\text{,}\) \(g(1) = 0\text{,}\) and \(f'(1) = g'(1)\text{.}\)
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Answer.
\(a = -3\text{,}\) \(b = 2\text{,}\) \(c = 1\)
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Subsection 2.2.13 Disguised Derivative Examples

Exercise Group 2.2.18. Limits as Disguised Derivatives.

Evaluate each limit by first converting it to a derivative at a particular \(x\)-value.
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(a)
\(\displaystyle\lim_{x \to 1} \frac{x^{50}-1}{x-1}\)
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Hint.
This is the definition of \(\frac{d}{dx}(x^{50})\big|_{x=1} = 50x^{49}\big|_{x=1}\text{.}\)
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Answer.
\(50\)
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(b)
\(\displaystyle\lim_{x \to -1} \frac{x^{2/9}-1}{x+1}\)
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Hint.
This is the definition of \(\frac{d}{dx}(x^{2/9})\big|_{x=-1} = \frac{2}{9}x^{-7/9}\big|_{x=-1}\text{.}\)
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Answer.
\(-\frac{2}{9}\)
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