Can someone help me with derivatives

Find the derivative of the function y = cos(cos(cos(x))).

Chain rule three times in a row!

let's take an arbitrary differentiable function f instead of cos ( just because f is easier to type that cos :) )

(f(f(f(x))))' = f ' (f(f(x))) (f(f(x))) ' = f ' (f(f(x))) f ' (f(x)) f ' (x)

if f = cos then f ' = -sin so this is:

(cos(cos(cos(x)))) ' = -sin(cos(cos(x))) * -sin(cos(x)) * -sin(x) = -sin(cos(cos(x))) * sin(cos(x)) * sin(x)

y = cos(cos(cos(x))) 's derivative is :


-sin(cos(cos(x)))*sin(cos(x))*sin(x)