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Problem 96 Hard Difficulty

Suppose $ y = f(x) $ is a curve that always lies above the $ x $-axis and never has a horizontal tangent, where $ f $ is differentiable everywhere. For what value of $ y $ is the rate of change of $ y^5 $ with respect to $ x $ eighty times the rate of change of $ y $ with respect to $ x? $

Answer

$y=2$

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Video Transcript

in this problem, we have a description of some function y equals f of X, and then we have some other function which I'm going to call G and it is white of the fifth Power. So that would be f of X to the fifth power. And then we have a description of the relationship between the derivative of G and the derivative of why So we want to find the point where the derivative of G is 80 times the derivative of why so we could write it like this. The derivative of G would be g prime of X and we want that to be 80 times the derivative of why, which would be f prime of X. So let's go ahead and find the derivative of G. So notice that G of X is a composite function. So we use the chain rule to get its derivative would bring down the five and then we raised F of X to the fourth. And then we multiply by the derivative of the inside, which is F prime of X. So let's substitute that for G Prime of X. Now notice we have a factor of F prime of X on both sides so we can cancel that. So now we're left with five times f of X to the fourth equals 80. Let's divide both sides by five so f of X to the fourth equal 16. Let's take the fourth root of both sides and we know that we're looking for a positive because we're told that f of X lies above the X axis so we don't want a negative value just positive. So f of X equals two. So what's the value of why we just found it? F of X means why? So why equals two at this point where the derivative is 80 times the original derivative?