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Derivatives of Inverse Functions
Inverse Function Review
One application of the chain rule is to compute the derivative of an inverse function. First, let's review the definition of an inverse function:
We say that the function is invertible on an interval [a, b] if there are no pairsin the interval such thatand . That means there are no two x-values that have the same y-value. That's important, because if two x-coordinates map to the same y-coordinate, the inverse function (working in reverse) would map a single x-coordinate to multiple y-coordinates. That doesn't make sense, because f(x) could have more than one resulting value!
We say thatis the inverse of an invertible functionon [a, b] if:
For example, the functionsandare inverses onsinceon that interval. Note that it works both ways -- the inverse function of the original function returns x, and the original function performed on the inverse ALSO returns x.
Taking The Derivative
So, how do we differentiate an inverse function? Recall the chain rule:
Applying this to the definition of an inverse function, we have:
Let's see how to apply this to real examples.
Letsoas above. Then, and applying the formula we have:
This agrees with the answer we would get from viewingas the polynomial function.
The functionis invertible on the interval, with inverse. We know that, so applying our formula we see that
We can check that , which means that