The Second-Order Derivative

The derivative $f'(x)$ is itself a function of $x$, so it can be differentiated again. The result is the second-order derivative (or simply second derivative):

It is found in two steps: differentiate once to get $\dfrac{dy}{dx}$, then differentiate that result again.

Worked illustration — $y = x^4$. First derivative $\dfrac{dy}{dx} = 4x^3$. Differentiating again, $\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}(4x^3) = 12x^2$.

Meaning. If the first derivative is a rate of change, the second derivative is the rate of change of the rate of change. In motion, if $y$ is position then $\dfrac{dy}{dx}$ is velocity and $\dfrac{d^2y}{dx^2}$ is acceleration. Its sign also describes the bending of a curve: $f''>0$ where the curve is concave up, $f''<0$ where it is concave down — a fact used later when locating maxima and minima.