Q:

differentiation using quotient rule

The quotient rule, i've understood as ${vdu - udv}\over v^2$
Differentiating $1 \over x+1$ it's ${(x+1)(0)-(1)(1)}\over (x+1)^2$ which equals $-1 \over (x+1)^2$
Differentiating $-x \over x+1$ it's ${(x+1)(-1)-(-x)(1)}\over (x+1)^2$ which equals $-1 \over (x+1)^2$
Why are these the same, or rather, what am i doing wrong here?

A:

$$\frac{-x}{x+1}=\frac{-x-1+1}{x+1}=-1+\frac{1}{x+1}$$
They differ by a constant, hence the derivative is the same.
Alternative working:
$$\frac{1}{x+1}-\frac{-x}{x+1}=\frac{x+1}{x+1}=1$$

