4.2.1 The Second Derivative

We get more information about the graph of a function if we differentiate the function twice. The derivative of a function $f\left(t\right)$ is a new function that we have denoted $f\prime \left(t\right)$ or

For example, if

then

This new function also has a derivative, which we may denote $f\prime \prime \left(t\right)$ or

For $f\left(t\right)=t^3-t,$ this new function is $6t\mathrm{.}$

It is common practice to discard the brackets in the second derivative notation and to replace the symbols `dd` with `d^2` and `dt dt` with `dt^2`. This yields the notation

For our example,

Example 1

Calculate the second derivative of the function `f` defined by

Solution   Using the Sum, Exponential, and Power Rules, we calculate the first derivative:

Then we differentiate again, using the same rules:

We don't have to stop with first and second derivatives — we could talk of third derivatives, fourth derivatives, and so on. For now, we have no need for these higher derivatives,

and we confine our attention to first and second derivatives. However, higher derivatives will play an important role much later in the course.

Second derivatives are not entirely new in our development — we just didn't point them out before. When we were discussing falling bodies, we let $s\left(t\right)$ represent the distance an object had fallen at time t. The velocity $v\left(t\right)$ was the derivative of $s\left(t\right)$, and the acceleration $a\left(t\right)$ was the derivative of $v\left(t\right)$. That is,