Differentiate the following functions w.r.t. x. : x x+1

Question

Differentiate the following functions w.r.t. x. : \begin{equation}
\frac{x}{x+1}
\end{equation}

✓

Answer

Let $y=\frac{x}{x+1}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x}{x+1}\right) \text { } \\
& =\frac{(x+1) \frac{\mathrm{d}}{\mathrm{d} x}(x)-x \frac{\mathrm{d}}{\mathrm{d} x}(x+1)}{(x+1)^2} \\
& =\frac{(x+1)(1)-x(1+0)}{(x+1)^2} \\
& =\frac{x+1-x}{(x+1)^2} \\
& =\frac{1}{(x+1)^2}
\end{aligned}
$

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