Unit content
Higher-order derivatives
A derivative is itself a function, so it can be differentiated again. The second derivative is
$$f^{\prime\prime}(x)=\frac{d^2f}{dx^2}$$
and more generally the $n$th derivative is written
$$f^{(n)}(x)$$
The second derivative measures how the first derivative changes. On a graph, its sign describes local concavity: $f^{\prime\prime}>0$ corresponds to slopes increasing, while $f^{\prime\prime}<0$ corresponds to slopes decreasing.
In motion, successive derivatives of position give velocity, acceleration and then jerk. Higher-order derivatives record progressively higher-order changes and provide the information used by Taylor approximations.