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Numerade Educator

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Problem 3 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 = \tan \pi x $

Answer

$=\pi \sec ^{2} \pi x$

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Video Transcript

here we have a composite function y equals tangent of Pi X. Now there could be parentheses around the pie X. They didn't write them in in the book, but it means the same thing and having the parentheses there, it might make it a little easier to think about this problem. So we want to identify the inside function and the outside function. Let's go with what's inside the parentheses as their inside function g of X so g of X equals pi times x and then the outer function would be f of X equals tangent of X. Now we're going to find the derivative Do I. D. X. So we start by taking the derivative of the outside function, and the derivative of tangent is C can't squared. So we have c can't squared of pi X Still what was inside the original. Then we multiply by the derivative of the outside function. Our excuse me, the inside function and the derivative of the inside function would be the derivative of pi times X, So that would be pie. Okay, so we have the derivative and now, to simplify the answer, we might decide just to take this pie and right in front of the other part. So we have pie times, See? Can't squared of pi X.