Derivatives from a graph If possible, evaluate the following d...

Question

Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'.

c. (f^-1)'(f(2))

Accepted Answer

Welcome back, everyone. Evaluate the following derivative using the graph given below. We want to evaluate the derivative of F inverse at the point F of 3, and we're given a graph which illustrates Y equals F of X and Y equals F of X. For this problem we're going to use one of the properties of derivatives. If we want to evaluate the derivative of F inverse. 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. Of a. And now in this case our point of interest is F of 3. So before we begin and identify a, well, we can already say that A is equal to F of 3, but we're going to extract a numerical value. It says F of 3. So this means that X equals 3, and we want to identify the Y value for. The function at that point. So if x equals 3, then y equals 5 based on the graph. So A equals 5, and then we can rearrange our equation. We can say that F inverse derivative at 0.5 is equal to 1 divided by F at a point of F inverse. A. 5. All that we have to do is simply understand how to read the part inside of Prey. It says that we have F inverse at. 5. How do we get this value if we're not given the F inverse on the coordinate plane? Well, essentially this is the X value. Or which. The parent function. Which is F of X is equal to 5, and this is because when we think about the relationship between the parent function and the inverts function, we're simply swapping the domain and the range, right? So what is the X value for which F of X is equal to 5? Well, we already know that if F of X is equal to 5, then The X value is equal to 3, right? This is the same point that we previously analyzed, so we can label that F of X is equal to 5 and X is equal to 3. So essentially this part gives us 3 and we can once again rewrite the equation as F inverse derivative at 0.5 is equal to 1 divided by F.3. And now our final step involves finding F at 0.3 and we're given the graph of F. So if we want to do that, we already know that X equals 3, and we can see that if X equals 3, then F is also equal to 3 because this is the Y value for F.3. And our final answer would be 1 divided by 3 or simply 1/3. And that's it. Thank you for watching.

Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
c. (f^-1)'(f(2))

Key Concepts

Inverse Function Theorem

Chain Rule

Evaluating Derivatives at Specific Points

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