Question

What are the first and second derivatives of `f(x)=ln(x^(2x+1) )`?

Answer 1

First derivative: `(dy)/(dx) =2ln(x)-1/x+2`

2nd derivative: `2/x + 1/(x^2)` Did not have time to do the `(d^2)/dx` properly so just gave the answer!

Explanation:

Given:`y=ln(x^(2x+1))`

Write as : `y=(2x-1)ln(x)`

Using standard form `(dy)/(dx)=v (du)/dx+u(dv)/(dx)`
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Let `u=2x-1 -> (du)/dx=2`
Let `v=ln(x) ->color(white)(...) (dv)/dx = 1/x`

Then`(dy)/(dx) = ln(x)(2)+(2x-1)(1/x)`

`(dy)/(dx) = 2ln(x)+(2x)/x-1/x`

`(dy)/(dx) =2ln(x)-1/x+2`