Question

What's the derivative of `f(x) = (1/3) (arctan(3x))^2`?

Answer 1

`f'(x)=(2arctan(3x))/(1+9x^2)`

Explanation:

The first issue is the squared function. Use the chain rule:

`f'(x)=2/3(arctan(3x))^1d/dx(arctan(3x))`

To differentiate the `arctan` function, use the chain rule again. The general rule for `arctan` functions is `d/dx(arctan(u))=(u')/(1+u^2)`. Here, `u=3x`, so

`f'(x)=2/3arctan(3x)*3/(1+9x^2)`

This simplifies to be

`f'(x)=(2arctan(3x))/(1+9x^2)`