Higher-Order Derivatives

The derivative $\dfrac{dy}{dx}$ of a function is itself a function of $x$, so it can be differentiated again. Differentiating the first derivative gives the second derivative, written

Differentiating once more gives the third derivative $\dfrac{d^3y}{dx^3}=f'''(x)$, and so on. Collectively these are the higher-order derivatives.

Meaning. If the first derivative measures the rate of change of $y$, the second derivative measures the rate of change of that rate — how the slope itself is changing. (In later chapters the sign of the second derivative distinguishes maxima from minima and tells whether a curve bends upward or downward.)

Method — finding the second derivative:

1. Differentiate $y$ once to obtain $\dfrac{dy}{dx}$; simplify it fully.
2. Differentiate that result again with respect to $x$ to obtain $\dfrac{d^2y}{dx^2}$.
3. Where a relation between $y$ and its derivatives is to be verified (e.g. "show that $\dfrac{d^2y}{dx^2}=4y$"), compute both sides and check they are equal.

Illustration. For $y = x^5$: $\dfrac{dy}{dx}=5x^4$, and $\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}(5x^4)=20x^3$.