Question

What is the derivative of this function `y=tan^-1(2x^4)`?

Answer 1

`(8x^3)/(4x^8+1)`

Explanation:

Aside from just applying the chain rule, one helpful way to perform this derivation may be to substitute a variable u to break the equation into separate pieces of a puzzle, like this:

We know the chain rule is
`(df(u))/dx=(df)/(dx)*(du)/(dx)`

We can also let u (in this case) = `2x^4`

Now just take the derivative of the function in two puzzle pieces:

We know `d/(du)` `tan^-1(u)` is just `d/(dx)(Tan^-1 (x)) = 1/(x^2+1)`

SO...

`d/(du)(tan^-1(u)) = 1/(u^2+1)`

and

`d/(dx) (2x^4) = 8x^3` (By the power rule)

From here, just substitute back `u=2x^4` to get `1/((2x^4)^2+1)*8x^3`

Which simplifies to... `(8x^3)/(4x^8+1)`

V'oila! Our puzzle is complete!