Implicit differentiation of an exponential with a function within a function

Question

$\frac{\mathrm d}{\mathrm dx}(y+1)^{x+1}$

I was wondering how to solve the question above, or approach this question.

I tried it this way, it was wrong.

The first way I made $y=(y+1)^{x+1}$ (1) and then took the natural log on both sides differentiated, and then subbed (1) in, but it was not correct. So I was wondering why a) am I wrong b) how to solve this

Thank you!

Answer 1

$$\begin{aligned}y&=(y+1)^{x+1}\\ \ln y&=(x+1)\ln (y+1)\\ \frac{y'}{y}&=\ln(y+1)+\frac{x+1}{y+1}\cdot y'\\ y'\left[\frac{1}{y}-\frac{x+1}{y+1}\right]&=\ln(y+1)\\ y'&=\ln(y+1)\cdot \left[\dfrac{y(y+1)}{y-x}\right]\end{aligned}$$

Now all you have to do is solve for $y'$. Can you proceed?