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Numerade Educator

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Problem 18 Easy Difficulty

Differentiate the function.

$ y = \ln (\csc x - \cot x) $

Answer

$f^{\prime}(x)=\csc x$

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Video Transcript

in this problem, we are learning how to take the derivatives of logarithmic functions in this case specifically using chain rule and additionally, we have triggered a metric functions here. So we're going to have to know the derivative of those to find the derivative of our entire function. So let's first review the function were given. We have y equals the natural log of the coast seeking of X minus co tangent of X. And what we need is why prime so gurgles have to apply chain role. This is a composition of functions, so chain role will be our best way to find the derivative. So why prime would be equal to one over the co secretive X minus cogent co tangent of X times Negative Costigan of Ex co tangent X Plus Co ck in squared of X. Now why did we do that? Well, the derivative of natural log is one over X, so we would put one over the input of our natural log and then we'll multiply by the derivative of that input so we would get y prime equals once we simplify the co ck in squared of X minus kosygin X Co tangent Ex all over the Kosygin of X minus the co tangent of X. And now we can factor out the co second term. We'll get the Costigan of x times, cosi gen x minus Qattan gen x all over Kosygin X minus Qattan gen x. Now this is really nice because this entire portion in the parentheses e is the same as the denominator. So they're just going to cancel, and we'll find that why Prime is just the co sick in the back. So I hope that this problem helped you understand how we can derive log rhythmic functions, especially using the chain role.