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

Question

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

b. (f^-1)'(3)

Accepted Answer

Welcome back, everyone. Evaluate the following derivative using the graph given below. We want to evaluate the derivative of the inverse. Of F at 0.4 and we're given the graph which contains y equals F of X and Y equals F of X. So basically the original function and is derivative. What we're going to do for this problem is define the following property if we want to find the derivative of the inverse of f. At a specific point A, this is going to be equal to 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 for this problem. Notice that if we use the methods of analogy, A is simply equal to 4. So we can rewrite this formula as F inverse at. Point A and specifically its derivative, right, so the derivative of F inverse at 0.4 is equal to 1 divided by. F at F inverse at 4. All that we have to do is understand how to read that point. So if we consider F inverse at 0.4 well, this is simply the X value. For which The original function F of X is equal to 4 because we're swapping the domain and range whenever it comes to the comparison of the original parent function and the inverse function, right? So our graph gives us y equals F of X. We're given that, and what we want to do is identify the X value for which F of X equals 4, right, which means that Y is equal to 4. So we are identifying Y equals 4, we're extending a horizontal line, and then if we go down, we can see that the X value for which F of X is equal to 4, is equal to 2. So this means that we can simplify our expression as F inverse derivative at. 4 is equal to 1 divided by f derivative at 2. Right, because this is the X value for which F of X is equal to 4. And now we can simplify for that we will need the derivative of F, and we're given the curve in red which represents y equals F of X. So we want to locate the 0.2 and we want to find the derivative of F at 0.2. We can clearly see that at x equals 2. The Y value is equal to 3, so we end up with 1 divided by 3, which is 1/3. And that's our final answer. Thank you for watching.

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
b. (f^-1)'(3)

Key Concepts

Inverse Function Theorem

Derivative of a Function

Graphical Interpretation of Derivatives

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