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Numerade Educator

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Problem 4 Easy Difficulty

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = \sin(\cot x) $

Answer

$\frac{d y}{d x}=-\csc ^{2} x \cdot \cos (\cot x)$

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Video Transcript

here we have a composite function and we're going to identify the inside function and the outside function, and we'll call the inside function G of X. So we see inside the parentheses we have co tangent of X, and then the outside function will call that f of X. So we see outside the parentheses we have signed. So the f of X, the outside function will be the sign of X. Now we're going to find the derivative D. Y. D X, and will use the chain rule which tells us to start by taking the derivative of the outside function. So we take the derivative of sign, which is co sign So we have co sign of co tangent X. We still have the inside in there. Now we multiply that by the derivative of the inside function and the derivative of co tangent X is negative coast. He can't squared X. So here we have our derivative and what we can do if we want to simplify it is just right. The negative Cosi can't squared X part first, so it's going to look like negative Cosi can't squared. X Times CO sign of co tangent X