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Numerade Educator

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Problem 47 Easy Difficulty

Use logarithmic differentiation to find the derivative of the function.

$ y = (\cos x)^x $

Answer

$(\cos x)^{x}(\ln \cos x-x \tan x)$

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Video Transcript

in this problem, we are learning how to use log rhythmic differentiation to find the derivative of a function. So we're given the function. Why equals the co sign of X rays to the X. Now, using log rhythmic differentiation is going to be very helpful in this case because we can simplify that exponents using log properties. So the first thing that we're going to do is take the natural log of each side so we would get the natural log of y equals the natural log of the co sin of X rays to the X. And now we can essentially factor out that exponents and put it in the front of our equation or of our expression. So we would get the natural log of y equals x times, the natural log of the coastline of X. And now this looks much easier to calculate, um, using product rule and also change rule. Instead of using a more complicated route so we would derive each side of our equation, we would get d natural log y d y times d y e x equals e d x d x times the natural log of the coastline of X and then you would have to use chain rule because of this composition of functions plus x times d natural log coastline X over DX and then we can simplify this to get one over. Why times d y d X equals one times the natural log of the coastline of X plus x times one over the cosine x times Negative sign X And now we can simplify to get a much nicer, um, equation, we would get one over. Why times do I d x times the natural log of the cosine x minus x times a tangent of X And then we would want to put why on the other side we multiply by Why To get our differential by itself we would have Dwight d X equals y times the natural log of the coast side of x minus x times a tangent of X. But we know why. Why is the function we started with? So we're just gonna substitute that back in. So we're going to get d Y d. X equals the coastline of X rays to the X times this entire quantity the natural log of the coastline of X minus X times the tangent of X and that is our derivative. I hope that this problem helped you understand how we can use log rhythmic differentiation to find the derivative of a function.