INVERSE FUNCTIONS
Theorem 7. The derivative of an inverse function is equal to
the reciprocal of the derivative of the direct function.
In the equations to this point, x has been the independent
variable and y has been the dependent variable. The equations have been in a
form such as
Suppose that we have a function like
and we wish to find the derivative
. Notice
that if we solve for y in terms of x, using the quadratic formula, we get the
more complicated function:
if we call this function the direct function, then
^{}
is the inverse function. To determine
from the inverse
function is easy.
EXAMPLE:
Find the derivative
of
the function
SOLUTION.The derivative
is
The reciprocal of
is the derivative
of the
direct function, and we find
EXAMPLE: Find the derivative
of the
function
SOL UTION.^{} Find
to be
Then
PRACTICE PROBLEMS:
Find the derivative
of the
following functions:
ANSWERS:
