The Second Derivative

The Derivative of a Derivative

Differentiation produces a new function, , from a function f(x). The second derivative of f(x) is simply the derivative of the derivative.

In addition to the prime notation, you will also frequently see the Leibniz notation for second derivatives:

Computing second derivatives introduces no new ideas, because the second derivative is simple the derivative of the derivative. The only practical problem with computing second derivatives is that sometimes the first derivative results in a messy expression. Differentiating that further can be messy and tedious.

The Second Derivative and Acceleration

The first derivative is a rate of change. For any function f(x) the second derivative is the rate of change of f(x) with respect to x. Continuing on in this vein, we see that the second derivative is the rate of change of the rate of change, or more commonly, the acceleration of f(x).

For example, if the position of an object in time is given by an expression

the rate of change of its position with respect to time, or its velocity, is given by

and the rate of change of its velocity, or its acceleration, is given by

Section 3.7 Homework

10, 11, 29, 30, 46