Question

How do you find the derivative of `cos(cos(cos(x)))`?

Answer 1

`-sin(x)sin(cos(x))sin(cos(cos(x)))`

Explanation:

It looks like the chain rule would be handy here. The chain rule states that for a function `f` where;

`f(x)=g(h(x))->f'(x)=g'(h(x))*h'(x)`

In the given expression, we have a lot of `cos` terms, and the derivative of `cos(x)` is `-sin(x)`, so applying the chain rule to our expression we get;

`d/(dx)cos(cos(cos(x)))=-sin(cos(cos(x)))*d/(dx)cos(cos(x))`

We still have `d/(dx)` but we can apply the chain rule again on `cos(cos(x))`.

`-sin(cos(cos(x)))(-sin(cos(x))*d/(dx)cos(x))`

One last time, we can take the derivative of `cos(x)`.

`-sin(cos(cos(x)))(-sin(cos(x))(-sin(x)))`

We can make this look slightly nicer by clearing out some of the parenthesis and negative signs.

`-sin(x)sin(cos(x))sin(cos(cos(x)))`

If there is a way to de-nest trig functions, I don't know it, so this is the answer.