Question

How do you find the derivative of `ln(x^2 +1)`?

Answer 1

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

Explanation:

You have a composed function `f(g(x))`, where `f(x)=ln(x)`, and `g(x)=x^2+1`

The rule for deriving composite functions is

`d/dx f(g(x)) = f'(g(x))*g'(x)`

which can be translated as "compute the derivative of the outer function with the inner function as argument, and multiply the derivative of the inner function".

To complete our scheme, we need the derivatives: we have

`f(x)=ln(x) \implies f'(x)=1/x`
`g(x)=x^2+1 \implies g'(x) =2x`

So, `f'(g(x)) = 1/g(x) = 1/(x^2+1)`, and the whole solution is `1/(x^2+1)*2x = (2x)/(x^2+1)`