This example demonstrates that a function and its derivative, , may have different domains.
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 , namelyThis 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.
Since the two one-sided limits are not equal it follows that Therefore, DOES NOT EXIST, which means that is NOT DIFFERENTIABLE at ! To summarize
and . Then, , and , for all real numbers .
So, the derivatives of these two different functions are equal.
Let’s compare the graphs of and for the derivatives we’ve computed so far: