Evaluate the following derivatives. d/(dx)((x+1)^(2x))

Step 1
To Determine: Evaluate the following derivatives.
Given: we have a function ${\displaystyle {(x+1)}^{2}x}$
Explanation, we have a function ${\displaystyle y={(x+1)}^{2}x}$ we have to find the derivative of y now we will applying exponent rule ${\displaystyle a^b=e^{b\ln \left(a\right)}sowehave{(x+1)}^{2}x=e^{2x\ln (x+1)}}$ now we will applying the chain rule
${\displaystyle \frac{d}{dx}\left(e^{2x\ln (x+1)}\right)=e^{2x\ln (x+1)}\frac{d}{dx}\left(2x\ln (x+1)\right)}$
Step 2
Now we will use product rule
${\displaystyle e^{2x\ln (x+1)}\frac{d}{dx}\left(2x\ln (x+1)\right)=e^{2x\ln (x+1)}\times 2\times [x\frac{d}{dx}\ln (x+1)+\ln (x+1)\frac{dx}{dx}]}$
${\displaystyle =2e^{2x\ln (x+1)}[\frac{x}{x+1}+\ln (x+1)]}$
${\displaystyle =2{(x+1)}^{2x}[\frac{x}{x+1}+\ln (x+1)]}$