The quotient rule is used to determine the derivative of a function expressed as the quotient of 2 differentiable functions. It is defined as shown:

Also written as:

This can also be done as a Product rule (with an inlaid Chain rule):

You may do this whichever way you prefer.

## Proof

We know that the two following limits exist as are differentiable. We also have the condition that .

Applying the first principles definition of differentiation we get

We can combine the two fractions into one fraction by cross-multiplying.

If we add and subtract a , it will change nothing.

If we bring the term in the denominator to the front, and separate the numerator with algebra of limits we will have

We now break this into a product of two limits, and a sum of two limits.

Evaluating all limits yields

Putting this in a more familiar form, we have

This completes the proof.