Proving Quotient Rule using Product Rule

This is how we can prove Quotient Rule using the Product Rule.

First, we need the Product Rule for differentiation: $\displaystyle\boxed{\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}}$

Now, we can write $\displaystyle\frac{d}{dx}(\frac{u}{v})=\frac{d}{dx}(uv^{-1})$

Using Product Rule, $\displaystyle \frac{d}{dx}(uv^{-1})=u(-v^{-2}\cdot\frac{dv}{dx})+v^{-1}\cdot(\frac{du}{dx})$

Simplifying the above will give the Quotient Rule! :

$\displaystyle\boxed{\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}$

You can also try proving Product Rule using Quotient Rule!