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Numerade Educator

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Problem 44 Hard Difficulty

Find the derivative of the function.
$ y = 2^{3^{4^{x}}} $

Answer

$\ln 2 \ln 3 \ln 4 \cdot 2^{3^{x}} \cdot 3^{4^{x}} \cdot 4^{x}$

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Video Transcript

okay, I think someone had a lot of fun writing this function, and it helps to stop and remember how to find the derivative of an exponential function, for example, how to find the derivative of two to the X. That would be to to the X Times natural log to. So what if it's more complicated, like to to the f of X? Well, then the chain rule comes in. You would have to to the f of x times, natural log to and that would be the derivative of the outside times f prime of X, the derivative of the insight. So let's see if we can apply that to this problem. So we have to tow a power. So the derivatives going to be two to that power times natural log to times the derivative of that power. So now we need the derivative of three to the four to the X, so again used the exponential function idea, but with three to the X instead of two to the X. So the derivative of three to the four to the X will be three to the four to the X Times natural log three times the derivative of four to the X. So what's the derivative of force? The X? Well, that would be four to the X Times natural log for okay. Now we want to take a look and see if there's anything we could do to make this a little bit simpler. So I suppose if we wanted to, we could rearrange the order of things and kind of group things together. So we have white prime equals Natural log two times natural log three times natural log four times, two to the three to the four to the X times three to the four to the X Times four to the X. That makes the answer look a little bit more fun.