The Product Rule

Most of the effort involved in differential calculus goes to finding the derivative function of some function that has been given to us. In order to do this for the broadest set of differentiable functions, we need to be equipped with a small set of tools, or methods for doing said differentiation. Here we develop the method used to differentiate the product of two functions e.g. d/dx [f(x)⋅g(x)]. This method is called, oddly enough, the product rule. We’ll see that this method necessarily differs from differentiating the sum of two functions, I’ll give an example that demonstrates how to apply the product rule, explain how we arrive at it, and how the quotient rule can be derived from it. If you’d like a sort list of the most common rules used to take derivatives of various types of functions you can find that here. Before launching into the product rule, however, here’s a quick review of the power and sum rules.

Fig. 1 The functions used to describe velocity and acceleration (and jerk) can be derived from an object’s position function (i.e. the function that describes how an object’s position changes over time). Functions of these types can be differentiated using a combination of the power and sum rules. Most functions that describe other types of behavior require us to know additional methods of derivation. In this article, we’ll examine the product and quotient rules.

The power and sum rules

Typically, the first method learned is the power rule which allows us to differentiate a term raised to a power. That is:

combined with the sum rule, which tells us that derivatives can be added together: