Higher Order Derivatives

A higher order derivative is simply the derivative of a derivative: you differentiate, then differentiate the answer again. The second derivative is the rate of change of the rate of change, and the most familiar example is acceleration (how fast your velocity is changing). The notation $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$ looks heavy, but it's just $\frac{dy}{dx}$ (the first derivative) with $\frac{d}{dx}$ (take the derivative again) applied to it. No new rules at all; you just use the old ones twice.

Remember that $f'(x)$ is itself a function, with its own graph and its own slopes. So there's nothing stopping us from differentiating it. The result, $f''(x)$, tells you whether the slope of $f$ is increasing or decreasing. When $f''$ is positive, the slope is going up and the graph bends upward like a cup. When $f''$ is negative, it bends downward like a cap. We'll use that a lot in the next unit to sketch graphs.

Notation. $f'$, $f''$, $f'''$, then $f^{(4)}, f^{(5)}, \dots$, with parentheses so the number isn't mistaken for a power.

Motion. If $s(t)$ is position, $s'(t)$ is velocity and $s''(t)$ is acceleration. These two can be very different, and that's worth thinking about. Picture throwing a ball straight up. At the top, it's stopped for an instant, so its velocity is zero. But it's still accelerating, because gravity is changing its velocity from upward to downward. Zero velocity and nonzero acceleration at the same moment is completely normal: it happens every time something turns around.

Patterns.

• Polynomials eventually differentiate to 0: the $(n+1)$st derivative of a degree-$n$ polynomial is 0.
• $\frac{d^n}{dx^n}e^{kx} = k^n e^{kx}$.
• Sine and cosine cycle every four derivatives: $\sin \to \cos \to -\sin \to -\cos \to \sin$.

Acceleration is a second derivative, and so is the "jerk"-free motion planning used for elevators and robot arms, which keeps the third derivative small for comfort.