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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 , 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
Let . Using the alternate expression for the derivative, find the slope of the tangent
line to the curve at the point .
The slope of the tangent line is given by the
derivative, . Now substitute in for the function we know, Now expand the
numerator of the fraction, Now combine like-terms, Factor an from every term in
the numerator, Compute the limit,
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.
Given the function , find .
Start with the definition of Replace with its formula, Simplify the top, Evaluate
Given the function , find .
Recall, the domain of is and is in fact a piecewise defined function, since We
will first compute when . Start with the definition of Replace with its
formula, Note: When , then for all small enough values of it follows that .
Therefore, . Now we have Now, we can compute the derivative when . Replace
with its formula, Note: When , then for all small enough values of it
follows that . Therefore, . Now we have What remains to be done is to check
whether the derivative exists. Start with the definition of Replace with
its formula, Note: When , then , but when , then . Therefore, instead of
computing the limit above, we will compute the two one-sided limits and compare
Since the two one-sided limits are not equal it follows that Therefore,
DOES NOT EXIST, which means that is NOT DIFFERENTIABLE at ! To
Is it true that for any function the domain of is equal to the domain of ?
This is not true. Consider the function . The domain of is and the domain of is
This example demonstrates that a function and its derivative, , may have different
Can two different functions, say, and , have the same derivative?
Many different functions can share the same derivatives. Consider two different
functions, and , defined by 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
For each of the three pairs of functions, describe when is positive, and when is
When is positive, is positiveincreasingnegativedecreasing. When is negative, is positiveincreasingnegativedecreasing
Here we see the graph of , the derivative of some function .