One Rule to Divide Them All: The Quotient Rule

Now that we can find the derivatives of equations containing sums or products, it's time we move on to equations involving quotients. For instance, given function which represents how are we to find the derivative?  The Quotient Rule is designed for just such a purpose.  So, how do we find the rule?  Well, you could Bing it, or we can think through it using what we know about derivatives.

For starters, we need to remember that the derivative is a rate of change.  We can recall that



Looking at our original equation, , we can think about it as the quotionent of two functions and .  This means that .  Applying the definition of a derivative, we get

 That's pretty messy, so we're just going to think about the top part for now: .  Algebra being what it is, we can work some trickery up here then add it back into the full equation later.

The first thing to do is to combine the two fractions utilizing common denominators giving us:

Simple is usually better, but to make any progress, we need to add something to this equation.  Keeping in mind we're only looking at the top of the equation, we need to find a way to get the patterns and to show up in some meaning full way.  I know it's a vague goal, but it helps to explain why our next step is to add  and subtract (so that it zeroes out) the value .  Given this new addition and some rearranging, our equation now looks like this:



Simplifying, we get:
Now we factor out a from the left side and from the right side and we get:

It's now time to plug this back into our full equation which gives us:

Simplifying, we get:

Using some limit trickery, we know this equal to

Further limit work lets us pull out a part of the denominator on both sides because the limit of a product is the product of the limits:

We're left with two limits we can solve, and two that are very close to the difference equation we're looking for.  We'll solve the simple limits first to clean it up and we have:

We can now pull out part of the numerator on both sides using the same limit theorem. On the left we pull out  while on the right we pull out .  These likewise are easy to solve and making our full equation:

We can now clearly see the difference quotients in there and can reduce to or even more simply:


We have now added the Quotient Rule to our bag of tricks.  We know that when it means

Let's apply this to our original function:

This means that and .  While and . Using our new found rule, we know that