Here we study the derivative of a function, as a function, in its own right.

The derivative of a function, as a function

First, we have to find an alternate definition for , the derivative of a function at .

Let’s start with the average rate of change of the function as the input changes from to . We will introduce a new variable, , to denote the difference between and . That is or . Take a look at the figure below.

Now we can write What happens if ? In other words, what is the meaning of the limit Obviously, this limit represents , the instantaneous rate of change of at ! Therefore, we have an alternate way of writing the definition of the derivative at the point , namely

This alternate definition of the derivative of at , namely,

(provided that the limit exists), allows us to define for any value of ,

(provided that the limit exists).

And this is how we define a new function, , the derivative of . The domain of consists of all points in the domain of where the function is differentiable. gives us the instantaneous rate of change of at any point in the domain of .
Given a function from some set of real numbers to the real numbers, the derivative is also a function from some set of real numbers to the real numbers. Understanding the relationship between the functions and helps us understand any situation (real or imagined) involving changing values.

Is it true that for any function the domain of is equal to the domain of ?
yes no
Can two different functions, say, and , have the same derivative?
yes no

Let’s compare the graphs of and for the derivatives we’ve computed so far:

For each of the three pairs of functions, describe when is positive, and when is negative.

When is positive, is positiveincreasingnegativedecreasing . When is negative, is positiveincreasingnegativedecreasing

Here we see the graph of , the derivative of some function .

Which of the following graphs could be ?