Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

f(x) = In e^x^2

Answer 1

`(d)/(dx) (ln(e^(x^2)) ) = (1)/(e^(x^2)) 2x e^(x^2) = 2x`

Explanation:

For this case we want to find the derivate of this function:

`y = ln(e^(x^2)) )`

And in order to find the derivate we need to apply the chain rule given by:

`(df(u))/(dx) =(df)/(du) (du)/(dx)`

And on this case
`f = ln(u), u = e^(x^2)`

And we can find the partial derivates like this:

`(d)/(du) (ln(u)) =(1)/(u)`

`(d)/(dx)(e^(x^2))= e^(x^2) (2x)`

And if we replace we got:

`(d)/(dx) (ln(e^(x^2)) ) = (1)/(u) 2x e^(x^2)`

And if we replace
`u = e^(x^2)` we got:

`(d)/(dx) (ln(e^(x^2)) ) = (1)/(e^(x^2)) 2x e^(x^2) = 2x`

And that would be our final answer on this case