Quotient rule
To derive a quotient, use the quotient rule.
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Remember
$f(x)=\frac{g(x)}{h(x)}$
$f'(x)=\frac{\color{red}{g'(x)}\cdot h(x) -\color{red}{h'(x)}\cdot g(x)}{(h(x))^2}$
$f'(x)=\frac{\color{red}{g'(x)}\cdot h(x) -\color{red}{h'(x)}\cdot g(x)}{(h(x))^2}$
Example
$f(x)=\frac{x}{x+1}$
Split function into subfunctions
$g(x)=x$ and $h(x)=x+1$Derive subfunctions
$g'(x)=\color{blue}{1}$ and $h'(x)=\color{green}{1}$Insert
$f'(x)=\frac{\color{blue}{g'(x)}\cdot h(x) -\color{green}{h'(x)}\cdot g(x)}{(h(x))^2}$
$f'(x)=\frac{\color{blue}{1}\cdot (x+1) -\color{green}{1}\cdot x}{(x+1)^2}$ $=\frac{1}{(x+1)^2}$