Theorem 5.1.1.
For a differentiable function \(f\) and small \(\Delta x\text{,}\)
\begin{equation*}
\boxed{\frac{\Delta y}{\Delta x} \approx f'(x)} \qquad \text{and} \qquad \boxed{\Delta y \approx f'(x) \Delta x}
\end{equation*}
Section 5.1 Differentials, Local Linear Approximations
Subsection 5.1.1 Derivatives and Local Linearity
One fundamental property of functions \(f\) which are differentiable is that they are locally linear. Roughly, this means that around a point \(x\) where \(f\) is differentiable, \(f\) can be approximated by a linear function. Intuitively, this means that “zooming in” on the function result in the function looking as if it were linear. In particular, this linear function is the tangent line of \(f\) at \(x\text{.}\)
Subsection 5.1.2 Approximating Small Changes
Recall that if \(y = f(x)\) is differentiable, then,
\begin{equation*}
\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}
\end{equation*}
where \(\Delta x\) is an increment in \(x\text{,}\) and \(\Delta y = f(x + \Delta x) - f(x)\text{.}\) Then, if \(\Delta x\) is not taken in the limit to 0, but rather is a non-zero small number, then because \(f\) is differentiable, the ratio \(\frac{\Delta y}{\Delta x}\) will be approximately \(f'(x)\text{,}\) or,
\begin{equation*}
\frac{\Delta y}{\Delta x} \approx f'(x)
\end{equation*}
This reasoning was used in the opposite direction to define the derivative. That is, we take the ratio of \(\Delta y\) and \(\Delta x\) to approximate \(f'(x)\text{.}\) On the other hand, if \(f'(x)\) is known, then this approximation can be used in the other way to determine \(\Delta y\) from \(\Delta x\) and \(f'(x)\text{.}\) Multiplying both sides by \(\Delta x\text{,}\)
\begin{equation*}
\Delta y = f'(x) \Delta x
\end{equation*}
In summary,
Subsection 5.1.3 Differentials
Previously, we considered the expression \(\frac{dy}{dx}\) as a single symbol representing the derivative. It is also helpful to interpret \(\frac{dy}{dx}\) as the ratio of two very small real number quantities, called differentials.
Definition 5.1.2.
Let \(y = f(x)\text{.}\) Then, a differential of x, \(dx\text{,}\) is given by \(\,dx = \Delta x\text{,}\) and \(dy\) is the differential of y, given by \(\,dy = f'(x) \,dx\text{.}\)
Subsection 5.1.4 Local Linear Approximation
Using the approximation, if a point \(y = f(x)\) is known, we can approximate points near \(x\) using the approximation,
\begin{equation*}
\boxed{f(x + \Delta x) \approx f(x) + \,dy = f(x) + f'(x) \,dx}
\end{equation*}
If \(f\) is differentiable at a point \(x = a\text{,}\) then \(f\) is locally linear around \(a\text{,}\) that is, can be approximated with a line. In particular, it can be approximated with its tangent line,
\begin{equation*}
\boxed{f(x) \approx f(a) + f'(a)(x - a)}
\end{equation*}
This approximation works well for \(x \approx a\text{.}\) The approximation is called a linearization.
Definition 5.1.3.
Let \(f\) be differentiable at \(x = a\text{.}\) Then, the linearization (or linear approximation) of \(f\) at \(x = a\) is the function,
\begin{equation*}
\boxed{L(x) = f(a) + f'(a)(x - a)}
\end{equation*}
This is useful because linear functions are very easy to evaluate, whereas a function \(f\) may be difficult or more complicated to evaluate.
A linear approximation is just the tangent line of a function, but thought of as a function \(y = L(x)\text{,}\) rather than a line (say, \(y = mx + b\)).
We most commonly consider approximations of functions at \(x = 0\text{,}\) in which case the tangent line formula is,
\begin{equation*}
f(x) \approx f(0) + f'(0)x
\end{equation*}
Example 5.1.4. Sine.
For \(f(x) = \sin{x}\text{,}\) \(f'(x) = \cos{x}\text{,}\) and so \(f'(0) = \cos{0} = 1\text{.}\) Then, the tangent line approximation is,
\begin{align*}
\sin{x} \amp \approx \underbrace{\sin{0}}_{=0} + \underbrace{\cos{0}}_{=1} x\\
\amp = x
\end{align*}
Thus,
\begin{equation*}
\sin{x} \approx x \qquad \text{for } x \approx 0
\end{equation*}
Example 5.1.5. Cosine.
For \(f(x) = \cos{x}\text{,}\) \(f'(x) = -\sin{x}\text{,}\) and so the tangent line approximation is,
\begin{align*}
\cos{x} \amp \approx \cos{0} - \sin{0} x\\
\amp = 1
\end{align*}
Example 5.1.6. Exponential.
For \(f(x) = e^x\text{,}\) \(f'(x) = e^x\text{,}\) and so the tangent line approximation is,
\begin{align*}
e^x \amp \approx e^0 + e^0 x\\
\amp = 1 + x
\end{align*}
Thus,
\begin{equation*}
e^x \approx 1 + x \qquad \text{for } x \approx 0
\end{equation*}
Example 5.1.7. Logarithm.
For the logarithmic function \(\ln{x}\text{,}\) it is not defined for \(x = 0\text{.}\) Instead, we can center the linear approximation \(x = 1\text{.}\) Then, for \(f(x) = \ln{x}\text{,}\) \(f'(x) = \frac{1}{x}\text{,}\) and so,
\begin{equation*}
\end{equation*}
Thus,
\begin{equation*}
\ln{x} \approx x - 1 \qquad \text{for } x \approx 1
\end{equation*}
Example 5.1.8. Binomial approximation.
For the function \(f(x) = (1 + x)^r\) (for some \(r > 0\)), \(f'(x) = r (1 + x)^{r-1}\text{,}\) and so,
\begin{gather*}
(1 + x)^r \approx (1 + 0)^r + r (1 + 0)^{r-1} x\\
\approx 1 + rx
\end{gather*}
Thus,
\begin{equation*}
(1 + x)^r \approx 1 + rx \qquad \text{for } x \approx 0
\end{equation*}
This is called the binomial approximation.
Subsection 5.1.5 Remark About Differentials
Historically, differentials, and linear approximations, were quite useful for making approximations. However, in modern times, with computers, differentials have becomes less useful. Today, differentials are mostly a helpful conceptual idea.
