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Problem 45 Hard Difficulty
Find the derivative of the function.
$ y = \cos \sqrt {\sin (\tan \pi x)} $
Answer
$=\frac{-\pi \cos (\tan \pi x) \sec ^{2}(\pi x) \sin \sqrt{\sin (\tan \pi x)}}{2 \sqrt{\sin (\tan \pi x)}}$
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Video Transcript
we're going to find the derivative of this function, and I'm going to start by changing the notation just a little bit. I'm going to change the square root notation two exponents rotation. So we have Why equals co sign of the sign of tangent Pi X to the 1/2 power. Okay, so how many layers of chain do we have here? We have the inside most layer pi times x the next layer tangent the next layer sign the next layer 1/2 power and then the outermost layer co signs. We have five altogether. So we're going to be multiplying five derivatives together. So why Prime is the derivative of co sign the outermost one is negative sign. So have negative sign of everything that was inside it. I'm gonna go ahead and write it as a square root just for the sake of writing this so I don't have to change it later. Okay, Now we move on and multiply that by the derivative of the next layer. And the next layer was the 1/2 power. So we bring down the 1/2 and we raised the inside to the negative 1/2 the inside is sign of Tangent of Pi X So that's to the negative 1/2 power. Now we move to the next layer which is sign and the derivative of Sinus co sign. So we have co sign of Tangent of Pi X. Now we move to the next layer, which is tangent and the derivative of tangent is he can't squared So we have c can't squared of Pi X and then we moved to the innermost layer Pi x and it's derivative would be pie. Now let's see what we can do to simplify our answer. We have something with the negative exponents so we could bring that to the denominator. We can bring the two to the denominator from the 1/2 and we just have a single negative sign We can combine the negative sign and the pi toe lead us off So we have the opposite of pie times the sign of the square root of sign of tangent Pi X So we've taken care of the negative and the sign and the pie So far Now we're bringing the 22 the bottom from the 1/2 and then since we have a negative 1/2 power here. We'll bring that to the bottom and will make that a square root. Okay, so now we have the co sign of tangent Pi X on the top. And then we have the c can't squared of Pi X also on the top and nothing else combines. So that's it. That's our lovely answer.
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