Calculus of a Single Variable, Chapter 2, 2.4, Section 2.4, Problem 64

Given the function y=cos(sqrt(sin(tan(pi x)))) . We have to find the derivative.
Let us begin,
(dy)/(dx)=-sin(sqrt(sin(tan(pi x)))).d/(dx)[sqrt(sin(tan(pi x)))] ------>(1)

Now,
d/(dx)[sqrt(sin(tan(pi x)))]=1/(2sqrt(sin(tan(pi x)))).d/(dx)[sin(tan(pi x))] ----->(2)
Again,
d/(dx)[sin(tan(pi x))]=cos(tan(pi x))d/(dx)[tan(pi x)]-------->(3)
=cos(tan(pi x)). pi sec^2(pi x)
Now substituting this in (2) we get,
d/(dx)[sqrt(sin(tan(pi x)))]=1/(2sqrt(sin(tan(pi x)))).cos(tan(pi x)).pi sec^2(pi x)

Substituting this in (1) we get,
(dy)/(dx)[cos(sqrt(sin(tan(pi x))))]=-sin(sqrt(sin(tan(pi x)))).pi/(2sqrt(sin(tan(pi x)))).cos(tan(pi x))sec^2(pi x)