Question

What is the derivative of `arctan(6x)`?

Answer 1

`6/(1+36x^2)`

Explanation:

Recap that `d/dx arctan (x) = 1/(1 + x^2)`

By the chain rule, if `y` is a function of `u` and `u` is a function of `x`, then `dy/dx = dy/(du) * (du)/dx`

Let `u=6x \Rightarrow (du)/(dx) = 6`
`y=arctan(6x)=arctan(u) \Rightarrow dy/(du) = 1/(1+u^2)`

Therefore by the chain rule,
`dy/dx = dy/(du) * (du)/dx`
`d/dx (y) = 1/(1+u^2) * 6`

Re-substituting `u=6x` and `y=arctan(u)=arctan(6x)`:
`d/dx arctan(6x) = 1/(1+(6x)^2) * 6 = 6/(1+36x^2)`