# How can you derive the quotient rule?

Aug 19, 2014

This can be proven fairly quickly, assuming knowledge of prior subjects such as the product rule and chain rule. Suppose $f \left(x\right) = \frac{u \left(x\right)}{v \left(x\right)}$. As we know that all of our equations are in terms of $x$, henceforth $x$ will be omitted from the steps below. Note however that it is still present as the variable for the functions.

$\left(\frac{d}{\mathrm{dx}}\right) f = \left(\frac{d}{\mathrm{dx}}\right) \frac{u}{v}$

Then via our definition $f = \frac{u}{v}$ we get $u = f \cdot v$. Differentiating this via use of the product rule nets us...

$u ' = f ' \cdot v + f \cdot v '$

Now as we isolate f' on its own side...

$f ' = \frac{u ' - f \cdot v '}{v}$

Recalling that $f = \frac{u}{v}$ this becomes...

$f ' = \frac{u ' - \left(\frac{u}{v}\right) \cdot v '}{v}$

And by multiplying both the numerator and denominator by $v$ we get...

$f ' = \frac{u ' \cdot v - u \cdot v '}{{v}^{2}}$

Or, by showing $x$ again...

$f ' \left(x\right) = \frac{u ' \left(x\right) \cdot v \left(x\right) - u \left(x\right) \cdot v ' \left(x\right)}{v \left(x\right)} ^ 2$