Question

What is the derivative of `(arctan x)^3`?

Answer 1

`(dy)/(dx)=(3(arctanx)^2)/(1+x^2)`

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)`.

Here we have `y=(arctanx)^3`

Hence `(dy)/(dx)=d/(d(arctanx))(arctanx)^3xxd/(dx)(arctanx)`

= `3(arctanx)^2xx1/(1+x^2)`

= `(3(arctanx)^2)/(1+x^2)`