Proving Quotient Rule using Product Rule

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!

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