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Numerade Educator

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Problem 44 Easy Difficulty

Use logarithmic differentiation to find the derivative of the function.

$ y = x^{\cos x} $

Answer

$y^{\prime}=x^{\cos x}\left(\frac{\cos (x)}{x}-\sin (x) \ln (x)\right)$

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Video Transcript

So in math applications, we often use exponential functions, trigger trigger metric functions. So for that reason, it's helpful for us to be able to, um, essentially take the derivative of those when wearing calculus, the derivatives, such a crucial part of understanding rates of change and all those things that we want to be able to use logarithms exponents, trigger metric functions and be able to take their derivatives. So one way in which we do this is if we have a complicated function such as this one right here where we have, um, the exponential function, but the variables in both the base and the exponents, so only that we solve this is taking the natural log of both sides. When we do that, this coastline X is now power so it can move out in front and multiply. Then the natural log of why Now we can perform implicit differentiation. So what we'll have is the the D. X of the natural log of y is going to be won over. Why Times y prime and then uh co sign of X Natural times Natural. Lagerback's the derivative that will be co sign of X times one over X class Attrill log of acts times a negative sine x So when we combine, um, these right here we'll get the coastline of X over X. So let's because I an X over X Then we can get rid of all this. Um, and then what we'll have is the negative sign X times the natural log of X. So that will be a negative natural log of X times sine x. And then lastly, we want to multiply by. Why on both sides? Yeah, so out here will be why, but we know that why is X to the coastline X So that's we have our final answer. Um, like this. And we can even move the cynics depending on where we want to write it. But it ultimately means the same thing that's right here will be our final answer. Using implicit differentiation. Um, derivatives of logarithms s so on and so forth