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- Understanding the Product Rule for Derivatives
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## Understanding the Product Rule for Derivatives

When you begin to learn the concepts of calculus, start by learning how to take the derivatives of various functions. Learn that the derivative of sin(x) is cos(x), that the derivative of ax^n is anx^(n-1), and a number of other rules for basic functions you’ve seen through algebra and trigonometry. After learning about the derivatives for individual functions, look at the derivatives of the products of those functions, which dramatically expands the range of functions you can take the derivative of.

However, there is a large step in complexity when moving from taking derivatives of basic functions to taking derivatives of products of functions. Because of this big step in how complicated the process is, many students feel overwhelmed and have a lot of trouble really understanding the material. Unfortunately, many teachers don’t give students methods to address these issues, but we do! Let’s get started.

Suppose we have a function f(x) that consists of two regular functions multiplied together. We call these two functions a(x) and b(x), which would mean that we have f(x) = a(x) * b(x). Now we want to find the derivative of f(x), which we call f'(x). The derivative of f(x) will look like this:

f'(x) = a'(x) * b(x) + a(x) * b'(x)

This formula is what we call the product rule. This is more complicated than any previous formula for derivatives you will have seen up to this point in your calculation sequence. However, if you write every function you’re dealing with BEFORE you try to write f'(x), then your speed and accuracy will improve greatly. So the first step is to write down what a(x) is and what b(x) is. Then, in addition to this, find the derivatives a'(x) and b'(x). Once you’ve written all that, there’s nothing else to think about and just fill in the blanks for the product rule formula. That’s all there is to it.

Let’s use a difficult example to show how easy this process is. Suppose we want to find the derivative of the following:

f(x) = (5sin(x) + 4x³ – 16x)(3cos(x) – 2x² + 4x + 5)

Remember that the first step is to identify what a(x) and b(x) are. Clearly, a(x) = 5sin(x) + 4x³ – 16x ib(x) = 3cos(x) – 2x² + 4x + 5, since these are the two functions that are multiplied together to form f(x). Then, next to our paper, we just write:

a(x) = 5sin(x) + 4x³ – 16x

b(x) = 3cos(x) – 2x² + 4x + 5

With this written apart from each other, we now find the derivative of a(x) and ib(x) individually just below. Remember that these are basic functions, so we already know how to take their derivatives:

a'(x) = 5cos(x) + 12x² – 16

b'(x) = -3sin(x) – 4x + 4

With everything written down in an organized way, we don’t have to remember anything anymore! All work for this issue is complete. We just need to write these four functions in the correct order, which gives us the product rule.

Finally, write the basic form of the product rule, f'(x) = a'(x) * b(x) + a(x) * b'(x), and write the respective functions in place of a (x), a'(x), b(x) and b'(x). So back to where we are working our problem we have:

f'(x) = a'(x) * b(x) + a(x) * b'(x)

f'(x) = (5cos(x) + 12x² – 16) * (3cos(x) – 2x² + 4x + 5) + (5sin(x) + 4x³ – 16x) * (-3sin(x) – 4x + 4)

This is a very long derived function, but when we organize our thinking in an efficient way, we can take derivatives of products quickly and accurately, no matter how long the original function is!

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