The quotient rule proof can be quite tedious if we are only allowed to use the definition of the derivative. In this post, this presents an alternative way of proving the quotient rule if we are allowed to use the product rule.
The Quotient Rule:
Let be the quotient of differentiable functions. Then, we can rearrange so that we have
Assuming that is differentiable, we can use the product rule on , which yields
Then we isolate on one side of the equation.
Now, we substitute .
We have arrived at the desired result, and this completes our proof.