Question

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

Answer 1

`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)`