`F(x) = int_0^(e^(2x)) ln(t + 1) dt` Find the derivative.

`F(x) =int_0^(e^(2x)) ln(t+1)dt`

`F'(x)=?`

Take note that if the function has a form

`F(x) = int_a^(u(x)) f(t)dt`

its derivative is

`F'(x)=f(u(x))*u'(x)`

Applying this formula, the derivative of the function

`F(x) =int_0^(e^(2x)) ln(t+1)dt`

will be:

`F'(x) = ln(e^(2x)+1)*(e^(2x))'`

`F'(x)=ln(e^(2x)+1)*e^(2x)*2`

`F'(x)=2e^(2x)ln(e^(2x)+1)`

Therefore, the derivative of the given function is `F'(x)=2e^(2x)ln(e^(2x)+1)` .