Higher-Order Derivatives
1. Motivation
The derivative $f’(x)$ is itself a function of $x$, so nothing stops us differentiating it again. Physically, if $f(t)$ is position, $f’(t)$ is velocity, and differentiating once more gives acceleration — the rate of change of the rate of change. In mathematical biology, second derivatives distinguish accelerating growth from decelerating growth even when the first derivative is positive in both cases.
2. Definition
Definition 2.1 (nth derivative) Let $f$ be differentiable on an interval. If $f’$ is itself differentiable, its derivative is the second derivative of $f$,
\[\begin{equation} f''(x) = \frac{d}{dx}\left[f'(x)\right] = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}. \label{eq:second-derivative} \end{equation}\]More generally, the $n$th derivative $f^{(n)}(x)$ is obtained by differentiating $f$ a total of $n$ times, provided each intermediate derivative exists.
Remark Common notations for higher-order derivatives: $f’‘(x)$, $f’’‘(x)$, $f^{(n)}(x)$ for $n \geq 4$; equivalently $\dfrac{d^2y}{dx^2}, \dfrac{d^3y}{dx^3}, \dfrac{d^ny}{dx^n}$ in Leibniz notation, or $\ddot{f}$ for a second time-derivative.
3. Geometric and physical interpretation
Interpretation $f’(x) > 0$ means $f$ is increasing; $f’‘(x) > 0$ means the rate of increase is itself increasing — the curve bends upward (convex). Conversely $f’‘(x) < 0$ indicates the curve bends downward (concave). Points where $f’‘(x) = 0$ and concavity switches sign are inflection points.
Example 2.2 Logistic growth $f(t) = \dfrac{K}{1+e^{-rt}}$ has $f’‘(t) > 0$ for small $t$ (accelerating growth) and $f’‘(t) < 0$ as $f(t) \to K$ (decelerating growth), with the sign change marking the inflection point at $f(t) = K/2$ — the fastest-growth moment.
4. Differentiating repeatedly
Theorem 2.3 (nth derivative of a power)
\[\frac{d^n}{dx^n}\left[x^m\right] = \frac{m!}{(m-n)!}\,x^{m-n}, \quad n \leq m,\]and $\dfrac{d^n}{dx^n}\left[x^m\right] = 0$ for $n > m$ (any integer $m$).
Theorem 2.4 (Leibniz Rule) For a product of two functions,
\[\begin{equation} \frac{d^n}{dx^n}\left[f(x)g(x)\right] = \sum_{k=0}^{n} \binom{n}{k} f^{(k)}(x)\, g^{(n-k)}(x). \label{eq:leibniz} \end{equation}\]This generalises the product rule (Theorem 1.3, the $n=1$ case) to arbitrarily many differentiations, weighting terms by binomial coefficients.
Check against the ordinary product rule
Setting $n=1$ in \eqref{eq:leibniz} gives $\binom{1}{0}f g’ + \binom{1}{1}f’g = fg’ + f’g$, recovering Theorem 1.3 exactly.
5. Periodicity under repeated differentiation
Example 2.5 Trigonometric functions cycle every four derivatives:
\[\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \to \cdots\]so $\dfrac{d^n}{dx^n}[\sin x]$ depends only on $n \bmod 4$.
Worked case: 10th derivative of sin(x)
$10 \bmod 4 = 2$, matching the second entry in the cycle, so $\dfrac{d^{10}}{dx^{10}}[\sin x] = -\sin x$.
Example 2.6 For $f(x) = e^{kx}$, every derivative reproduces the original function up to a power of $k$:
\[f^{(n)}(x) = k^n e^{kx}.\]Show by induction
Base case $n=1$: $f’(x) = ke^{kx}$, matching Theorem 1.7. Inductive step: if $f^{(n)}(x) = k^n e^{kx}$, differentiating once more (Theorem 1.7 again) gives $f^{(n+1)}(x) = k^n \cdot k\, e^{kx} = k^{n+1}e^{kx}$.
Remark Higher-order derivatives underpin Taylor expansions (Lesson 3) and appear directly in PDE models throughout mathematical biology — e.g. the diffusion term $\partial^2 u/\partial x^2$ in reaction-diffusion equations is a second spatial derivative.