Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Fin...

Question

Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).

Accepted Answer

Hello there. Today we're gonna solve the fine practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given a function H of X and its inverse H to the power of -1 X. If the slope of Y equals H the power of -1 of X, at the point 5.1 is 2, what is H of -1? Awesome. So ultimately it appears the final answer and we're ultimately trying to solve for is we're trying to figure out what is H of -1, given the conditions set to us by the prim itself. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 1/2, B is negative 12, C is 2, and D is negative 2. OK, so first off, in order to find H to the power of -1 of 1, we need to use the relationship between the derivative of a function and its inverse, which, as we should recall, we need to note that H to the power of -1 X is equal to 1 divided by H Y. Where H of Y is equal to X, and let us note that we are given, and let's make a note of this in blue, let us note that we are given the slope, which the slope states that Y is equal to H to the power of -1. Of X At the points 51. Is 2. And we need to note that at this point X equals 5 and Y equals -1. Fantastic. So now our first step towards solving this problem is we need to use the form we need to recall and use the formula for the inverse derivative. So we need to take H H to the power of -1. when you take H to the power of -1. Of 5, Which is equal to 1 divided by H of -1. And we need to note that we are given, so let's make a note here that we are given. H the power of -1 of 5 is equal to 2, so we could go ahead and write that 2 is equal to 1 divided by H of -1, fantastic. So now, our second and final step is we need to solve for H of -1, so we're solving for this now, and what we need to do is we need to rearrange the equation. So when we rearrange the equation, we will find that H of -1 is equal to 1/2. And all we're doing is we're taking our expression. Which is 2 is equal to 1 divided by H of -1, and we're rearranging this expression to isolate and solve for H of -1, and when we do that, we will get an answer of 12, and that is our final answer. Hooray, we did it, and this corresponds with multiple choice answer letter A, which once again is one half. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).

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