Question

What is the derivative of `cos(tan x)`?

Answer 1

`(dy)/(dx)=-sin(tanx)sec^2x`

Explanation:

Chain Rule - 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)`

Here we have `y=cosu`, where `u=tanx`

Hence, `(dy)/(dx)=(dy)/(du)xx(du)/(dx)`

= `d/(du)cosuxxd/dx(tanx)`

= `-sinuxxsec^2x=-sin(tanx)sec^2x`