Derivatives: The Product Rule

Up to this point, we have learned a lot about derivatives.  We know you to find them, multiple times, and how to apply these rules in a large function using addition and subtraction.  Now, lets talk about a different situation.  If you are given this function:

f(x) = 5x⁴ * 2x

What do you do?  The logical answer would be the same way we differentiated functions with addition and subtraction.  However, its not quite that simple.  In order to differentiate this function, we have to use something called the product rule.

The product rule looks something like this:

f(x) = g(x) * h(x)
f '(x) = [g(x) * h'(x)] + [h(x) * g'(x)]

So, in words, what happens here is you have the first argument times the derivative of the second, added to the second times the derivative of the first.  It may sound confusing to begin with, but you'll get the hang of it quickly.  Some people may prefer to flip-flop the addends in this equation.  That is completely fine given the standard rules of addition.  As long as you are multiplying one times the derivative of the other, you're fine.  Now back to the example.  Given what we have just learned, lets put that rule to use.

f(x) = 5x⁴ * 2x
f '(x) = 5x⁴(2) + 2x(20x³)
f '(x) = 10x⁴ + 40x⁴

As you will see, I have stopped showing the simple steps of the derivative.  At this point, I am assuming you're a champ at differentiation so I won't keep showing you the steps.  If you have a question about one, however, feel free to ask or refer to the prior pages about simple derivatives.  I hope this gives you the basic understanding of finding derivative using the product rule.  Just remember, first times the derivative of the second plus the second times the derivative of the first.  A lot of math and memorization just takes practice, so do a few problems and you'll get the hang of it.  Next post, we will discuss the missing operand thus far:  division.