If possible, evaluate the following derivatives using the grap...

Question

If possible, evaluate the following derivatives using the graphs of f and f'.

a. (f^-1)'(7)

Accepted Answer

Welcome back, everyone, to another video. Evaluate the following derivative using the graph given below. We want to evaluate the derivative of the inverse function of F at 0.5, and we're given a graph. So for this problem we want to recall the property. If we want to take the inverse function of F and we want to evaluate its derivative at a point A, well, essentially what we have to do is take 1 divided by the derivative of the original function F at a point of F inverse at a, and this is the property that we want to use. So what we're going to do is simply understand what that point is, right? So essentially, our point of interest A is equal to 5, and what we can do is simply rewrite it as F inverse. The derivative of that at 0.5 is equal to 1 divided by. The derivative of F at. F inverse 8.5. And this tells that the point of interest is 5, but let's understand that we are not given the inverse function graph. We're given Y equals F of X and Y equals F of X. So we have to understand that if we're looking at F inverse at 0.5, we can understand that it is going to have a specific value which corresponds to the X value. When F of X, the original function of that inverse is equal to 5. So what we're going to do is identify the graph of Y equals F of X, and we want to see where it is equal to 5, right? So now, let's extend a horizontal line until we find an intersection point. And we can see that the x value when F of X is equal to 5 is equal to 3, right? And essentially we can say that our denominator is F at 3 because the inverts at 0.5. Is equal to we right we are flipping the domain in range when it comes to the comparison of the parent function in its inverse. So we are simplifying F inverse at 0.5 into simply 3, right, and our numerator still contains 1. So what we can do is continue and say that we have 1 divided by. And now we want to identify the derivative of F at 0.3. And if we carefully look at the given graph while we have the derivative Y equals F of X, let's notice that at 0.3 we have a value of 3 at the y axis as well. So F.3 is equal to 3, meaning the final answer to this problem corresponds to 1/3, and that would be the final answer to this problem. Thank you for watching.

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
a. (f^-1)'(7)

Key Concepts

Inverse Function Theorem

Derivative Interpretation

Graphical Analysis of Functions

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