Question

derivative of y = cos(sqrt(sin(tan pi x))) ?

Answer 1

Did your teacher really give you this problem? What a pain in the neck (hint the word I was gonna put here starts with an a). But it is good practice of...
Chain rule:
d/du(u) = 1
dx(tan(u)) = sec^2(u) * du/dx
d/dx of sin(u) = cos(u) * du/dx
d/dx(sqrt(u)) = 1/2u^(-1/2) * du/dx
d/dx(cos(u)) = -sin(u) * du/dx

Therefore:
\[\frac{d}{dx}(\cos(\sqrt(\sin(\tan (\pi x)))))) = -\sin(\sqrt(\sin(\tan (\pi x)))) * \frac{1}2[\sin(\tan (\pi x)))]^{-1/2}\]
\[* \cos(\tan (\pi x)) * \sec^2(\pi x) * \pi\]

I had to break it into two lines because it was so long that it wouldn't fit on one line lol

Answer 2

what a brilliant response @misha