# How do you use the quotient rule to differentiate 1 / (1 + x²)?

Feb 26, 2017

The quotient rule states that the derivative of some function that's expressed as a quotient of two other functions, such as if $f \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)}$, then the derivative of $f$ is given through:

$f ' \left(x\right) = \frac{g ' \left(x\right) h \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2$

For $f \left(x\right) = \frac{1}{1 + {x}^{2}}$, we see that $g \left(x\right) = 1$ and $h \left(x\right) = 1 + {x}^{2}$.

We then see that $g ' \left(x\right) = 0$ and $h ' \left(x\right) = 2 x$. Plugging these in gives:

$f ' \left(x\right) = \frac{0 \left(1 + {x}^{2}\right) - 1 \left(2 x\right)}{1 + {x}^{2}} ^ 2$

$f ' \left(x\right) = \frac{- 2 x}{1 + {x}^{2}} ^ 2$

$\text{ }$

Footnote

If you've learned the chain rule, it's easier to do this by rewriting the function as ${\left(1 + {x}^{2}\right)}^{-} 1$ and then seeing that the derivative is $- {\left(1 + {x}^{2}\right)}^{-} 2 \frac{d}{\mathrm{dx}} \left({x}^{2}\right)$, identical to what we found.