Higher-Order Derivatives

By the end of this lesson, you will be able to compute higher-order derivatives, interpret them physically (velocity, acceleration, jerk), and find patterns in higher-order derivatives of elementary functions.

Notation

$$f''(x) = \frac{d^2y}{dx^2} = \frac{d^2f}{dx^2} = D^2f$$

$$f^{(n)}(x) = \frac{d^n y}{dx^n} \quad \text{(}n\text{th derivative)}$$

Physical Interpretation

For position $s(t)$:

• $v(t) = s'(t)$: velocity (rate of change of position)
• $a(t) = v'(t) = s''(t)$: acceleration (rate of change of velocity)
• $j(t) = a'(t) = s'''(t)$: jerk (rate of change of acceleration)

Positive acceleration: speeding up in the positive direction (or slowing down in the negative direction).

Example 1 — Polynomial

$f(x) = 3x^5 - 2x^3 + x$

$f'(x) = 15x^4 - 6x^2 + 1$

$f''(x) = 60x^3 - 12x$

$f'''(x) = 180x^2 - 12$

$f^{(4)}(x) = 360x$

$f^{(5)}(x) = 360$

$f^{(6)}(x) = 0$

Degree-$n$ polynomial: $n$th derivative is constant, $(n+1)$th is zero.

Example 2 — Trigonometric Patterns

$$\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \to \cdots$$

Period 4: $(\sin x)^{(n)}$ depends on $n \mod 4$:

• $n \equiv 0$: $\sin x$
• $n \equiv 1$: $\cos x$
• $n \equiv 2$: $-\sin x$
• $n \equiv 3$: $-\cos x$

Similarly for $\cos x$ (offset by one).

Example 3 — Exponential

$\dfrac{d^n}{dx^n}[e^x] = e^x$ for all $n$.

$\dfrac{d^n}{dx^n}[e^{ax}] = a^n e^{ax}$

Example 4 — Physical Motion

$s(t) = t^3 - 6t^2 + 9t$:

$v(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3)$

$a(t) = 6t - 12$

At $t = 0$: $v = 9$ (moving positive), $a = -12$ (decelerating).

At $t = 1$: $v = 0$ (momentarily at rest), $a = -6$ (still decelerating).

At $t = 2$: $v = -3$ (moving negative), $a = 0$ (inflection in velocity).

Second Derivative and Concavity (Preview)

$f''(x) > 0$: $f$ is concave up (slope is increasing, bowl opening upward)

$f''(x) < 0$: $f$ is concave down (slope is decreasing, bowl opening downward)