A Quest for Calculus: 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: