Some function f(x) exists, and is continuous and differentiable along its entire domain. If you know that, and are given the selected values of f and f ' shown in the below table, evaluate . Then explain why the condition is necessary for this problem.

Difficulty:

Solution

First of all, you'll need to have a formula memorized, the formula for the derivative of an inverse function. In case you don't remember it, here it is:

To evaluate the derivative at x = 3, our first order of business is the set of innermost parentheses in the formula. In other words, we need to evaluate . Since we don't have values of the inverse function defined explicitly by the table, we have to be a little clever. Think of it like this: translates into "Where does f have an output of 3?". If you look at the chart, you see that . That information alone is enough to conclude that . Remember that the output for a function is the input for its inverse and vice versa.

Since we know that , let's plug that into the formula:

It's very simple to figure out what f '(4) is -- just look at the chart. It equals 5. So, plug that in to get the final answer. The derivative of the inverse function of f at x = 3 is .

It's important that because this guarantees that f is constantly increasing, and any function which is strictly monotonic (i.e. never changes direction) is one-to-one and therefore possesses an inverse function. If the function didn't have an inverse function, we wouldn't be able to find the derivative of the inverse function, like this question asks us to.