How do you differentiate y = cos(cos(cos(x)))?

1 Answer
Jul 12, 2016

dy/dx = -sin(cos(cos(x)))sin(cos(x))sin(x)

Explanation:

This is an initially daunting-looking problem, but in reality, with an understanding of the chain rule, it is quite simple.

We know that for a function of a function like f(g(x)), the chain rule tells us that:
d/dy f(g(x)) = f'(g(x)g'(x)

By applying this rule three times, we can actually determine a general rule for any function like this one where f(g(h(x))):
d/dy f(g(h(x))) = f'(g(h(x)))g'(h(x))h'(x)

So applying this rule, given that:
f(x) = g(x) = h(x) = cos(x)
thus
f'(x) = g(x) = h(x) = -sin(x)

yields the answer:
dy/dx = -sin(cos(cos(x)))sin(cos(x))sin(x)