Question

What is the derivative of `y = cos(cos(cos(x)))`?

Answer 1

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

Explanation:

In order to differentiate a function of a function, say `y, =f(g(x))`, where we have to find `(dy)/(dx)`, we need to do (a) substitute `u=g(x)`, which gives us `y=f(u)`. Then we need to use a formula called Chain Rule, which states that `(dy)/(dx)=(dy)/(du)xx(du)/(dx)`. In fact if we have something like `y=f(g(h(x)))`, we can have `(dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)`

Hence for `y=cos(cos(cos(x)))`

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

= `-sin(cos(cos(x)))xx(-sin(cosx))xxd/(dx)cos(x)`

= `-sin(cos(cos(x)))xx(-sin(cosx))xx-sin(x)`

= `-sin(cos(cos(x)))sin(cosx)sin(x)`