Question

How do you differentiate `g(x) = lnxe^(2x)` using the product rule?

Answer 1

`g'(x) = (2x+1)/x`

Explanation:

`g(x) = ln(xe^(2x))`

`g'(x) = 1/(xe^(2x)) * d/dx(xe^(2x))` Standerd differential and chain rule

`= 1/(xe^(2x)) * (x*e^(2x)*2 + e^(2x)*1)` Product rule, chain rule and standard differential

`=(e^(2x) (2x+1))/(xe^(2x))`

`=(cancel(e^(2x)) (2x+1))/(xcancel(e^(2x)))`

`= (2x+1)/x`