Section 3.5 Higher Derivatives
Subsection 3.5.1 Higher Derivatives
For a function \(f\text{,}\) its derivative \(f'\) is itself a function, and so it itself can be differentiated, to form a new function (the “derivative of the derivative”). This function is called the second derivative. The second derivative of \(f\) is denoted by \(f''\) (with two primes). In other words,
\begin{equation*}
f''(x) = \frac{d}{dx} f'(x)
\end{equation*}
Similarly,
- The third derivative is the derivative of the second derivative, and is denoted by \(f'''\text{.}\)\begin{equation*} f'''(x) = \frac{d}{dx} f''(x) \end{equation*}
- In general, the \(n\)th derivative is the function which results from differentiating \(f\) \(n\) times, and is denoted by \(f^{(n)}\text{.}\)
Then, the derivatives of \(f\) are denoted by,
\begin{equation*}
f', f'', f''', f^{(4)}, f^{(5)}, \dots, f^{(n)}, \dots
\end{equation*}
Typically, a number superscript is used for \(n \geq 4\text{.}\)
Or, in Leibniz notation, the higher derivatives are denoted by,
\begin{equation*}
\brac{\frac{dy}{dx}} \quad, \quad \frac{d^2 y}{dx^2}, \frac{d^3 y}{dx^3}, \frac{d^4 y}{dx^4}, \dots, \frac{d^n y}{dx^n}, \dots
\end{equation*}
Subsection 3.5.2 Intuition for Higher Derivatives
Given a function \(f\text{,}\) recall that the derivative of \(f\) can be thought of as the rate of change of \(f\text{.}\) Then, the derivative \(f''\) of \(f'\) can be thought of as the rate of change of \(f'\text{,}\) that is, measuring how fast the derivative changes. In terms of \(f\text{,}\) \(f''\) can be thought of as how fast the slope of \(f\) is changing.
Subsection 3.5.3 Intuition for Leibniz Notation for Higher Derivatives
Intuitively, the second derivative of \(y\) is the result of applying the derivative operator \(\frac{d}{dx}\) to \(y\) twice. In other words,
\begin{equation*}
y'' = \frac{d}{dx} \frac{d}{dx} y
\end{equation*}
Then, in some sense, the operators can be “combined” as,
\begin{equation*}
y'' = \brac{\frac{d}{dx}}^2 y
\end{equation*}
Or,
\begin{equation*}
y'' = \frac{d^2}{(dx)^2} y = \frac{d^2 y}{(dx)^2}
\end{equation*}
Finally, Leibniz notation writes \(dx^2\) rather than \((dx)^2\text{,}\) omitting the brackets but maintaining the intuition that the denominator is \((dx)^2\) rather than \(d(x^2)\text{.}\)
Subsection 3.5.4 Kinematics and Higher Derivatives
Recall that if \(x(t)\) is the position of an object at time \(t\text{,}\) its velocity and acceleration are given by \(v(t) = x'(t)\) and \(a(t) = v'(t)\text{,}\) respectively. Then, using the language of higher derivatives,
\begin{align*}
v(t) \amp = \frac{dx}{dt} \quad \text{is the first derivative of position}\\
a(t) \amp = \frac{dv}{dt} = \frac{d^2x}{dt^2} \quad \text{is the second derivative of position}
\end{align*}
