We can verify this by finding the inverse of the function first and then taking the derivative. The inverse of
the function f(x) = x^2 is the function f−1(x) = . Now we find the derivative and evaluate it at f(3) = 9.
Remember the rule: Find the value, a, of f(x) that gives you the value of x that the problem asks for. Then
plug that value, a, into the reciprocal of the derivative of the inverse function.
Example 2: Find a derivative of the inverse of y = x^3 − 1 when y = 7.
First, we need to find the x-value that corresponds to y = 7. A little algebra tells us that this is x = 2. Then,
= 3x^2 and Therefore, the derivative of the inverse is
Verify it: The inverse of the function y = x^3 − 1 is the function y = . The derivative of this latter
function is
Let’s do one more.
Example 3: Find a derivative of the inverse of y = x^2 + 4 when y = 29.
At y = 29, x = 5, the derivative of the function is