Question

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

Answer 1

`3/(1+x^2)(arctan(x))^2`

Explanation:

Note:
`color(white)("XXX")(d (arctan(x)))/(dx)=1/(1+x^2)`
and
`color(white)("XXX")(d(x^3))/(dx)=3x^2`

Using the chain rule:
`color(white)("XXX")(d(color(blue)((color(red)(arctan(x)))^3)))/(d(color(green)x))=(d(color(blue)((color(red)(arctan(x)))^3)))/(d(color(red)(arctan(x)))) * (d(color(red)(arctan(color(green)x))))/(dcolor(green)x)`

`color(white)("XXXXXXXXXXXXX")=3(arctan(x))^2 * 1/(1+x^2)`