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Problem 48 Easy Difficulty

Use logarithmic differentiation to find the derivative of the function.

$ y = (\sin x)^{\ln x} $

Answer

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

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Video Transcript

in this problem, we are learning how to find the derivative of a function using log rhythmic differentiation. And this is very helpful because we can simplify exponents very easily using log log properties. Excuse me. So we have the function. Why equal society of X rays to the natural log of X and this looks really complicated, but it's not going to be extremely complicated if we use log rhythmic differentiation. So the first thing we're going to do is take the natural log of each side. You would get the natural log y equals the natural log of the sign of X rays to the natural log of X. But using the properties of logs, we can pull out that exponents so we would get the natural log of y equals the natural log of X are exponents times the natural log of the sign of X, basically what's left over. So then we can derive each side of the equation. And for our right side, we have to use product rule as well as chain role. So we're going to get one over. Why times do I d X equals one over x times the knock tra log of sine X plus the natural log of X times. Now this is our changeable component one over the sine of x times, the co sign of X. And then we can simplify a little bit and we can move this. Why to the other side. So will multiply by Why and I did a little bit of simplification with our trig and metric function here will get d y d X equals y times one over x times a natural log of the sign of X plus the natural log of x times the co tangent of X. But we know what? Why is that? That was what what we started with. So we can just substitute that function in and we'll find that d Y d X equals the sign of x rays to the natural log of X times this entire quantity the natural log of the sign of X over X plus the natural log of x times, the co tangent of X and that is our derivative. So I hope this problem helped you understand a little bit more about the motivation behind log rhythmic differentiation and how we go through the process of applying it to a function and finding its derivative

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