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Numerade Educator

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Problem 33 Hard Difficulty

Find the derivative of the function.
$ G(x) = 4^{C/x} $

Answer

$=-C(\ln 4) \frac{4^{C / x}}{x^{2}}$

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Video Transcript

in this problem, we have an exponential function. So as a reminder, if you want to take the derivative of an exponential function like four to the X, it's going to be four to the X Times natural log. Four. So then, if we make it a composite function and we have four race to a function, it's going to be four. Race to that function times Natural log four. That's the derivative of the outside, multiplied by F prime of X. That's the derivative of the inside. So for this function will follow that rule. And first I'm going to rewrite it as four raised to the power See X to the negative one. Any time I have something divided by X, I think of it as X to the negative first power. Okay, so for the derivative, according to this exponential function derivative rule, we would have four race to the sea X to the negative one multiplied by natural log four multiplied by the derivative of C X to the negative one. And that would be negative one time See X to the negative too. So now what we need to do is just clean this up a little bit. So when I write four c x to the negative one, I'm gonna put that back into the fraction notation four to the sea over X Power times Natural log for times negative. See, In fact, I might just write that negative See in the front of my answer as we typically put a constant in the front and then we'll take this X to the negative. Second power will bring it down to the bottom as X squared.