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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 $f'(a)$, the derivative of a function $f$ at $a$.

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

Now we can write What happens if $h\to 0$? In other words, what is the meaning of the limit Obviously, this limit represents $f'(a)$, the instantaneous rate of change of $f$ at $a$! Therefore, we have an alternate way of writing the definition of the derivative at the point $a$, namely

This alternate definition of the derivative of $f$ at $a$, namely,

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

(provided that the limit exists).

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

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

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

For each of the three pairs of functions, describe $y=f(x)$ when $f'$ is positive, and when $f'$ is negative.

When $f'$ is positive, $y=f(x)$ is positiveincreasingnegativedecreasing . When $f'$ is negative, $y=f(x)$ is positiveincreasingnegativedecreasing

Here we see the graph of $f'$, the derivative of some function $f$.

Which of the following graphs could be $y = f(x)$?   