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## Understanding the Product Rule for Derivatives

When you start learning the concepts of differential calculus, you start by learning how to take the derivatives of different functions. You learned that the derivative of sin(x) is cos(x), that the derivative of ax^n is anx^(n-1), and many other rules for basic functions that you see in all algebra and trigonometry. After learning about the derivatives for individual functions, you will look at the derivatives of the products of these functions, which greatly expands the range of functions you can take the derivative of. in.

However, there is a big step in complexity when you move from taking derivatives of basic functions to taking derivatives of products of functions. Because of this huge step in how complicated the process is, many students feel overwhelmed and have a lot of problems really understanding the material. Unfortunately, many teachers do not provide students with 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. Let’s call these two functions a(x) and b(x), which means 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 of the previous formulas for derivatives you’ve seen up to this point in your calculus sequence. However, if you write down every function you deal 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 next to that, find the derivatives a'(x) and b'(x). If you have written everything, then there is nothing else to think about, and you fill in the blanks for the product rule formula. That’s all.

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. It is clear that a(x) = 5sin(x) + 4x³ – 16x and b(x) = 3cos(x) – 2x² + 4x + 5, since these are the two functions that are multiplied to form f(x) . On the side of our paper then, we just write:

a(x) = 5sin(x) + 4x³ – 16x

b(x) = 3cos(x) – 2x² + 4x + 5

With those written separately from each other, now we can find the derivatives of a(x) and b(x) individually under that. Remember that these are basic functions, so we already know how to get their derivatives:

a'(x) = 5cos(x) + 12x² – 16

b'(x) = -3sin(x) – 4x + 4

With everything written in an organized way, there’s no need to remember anymore! All work for this problem is done. We just need to write these four functions in the correct order, which is given to us by the product rule.

Finally, you write the basic form of the product rule, f'(x) = a'(x) * b(x) + a(x) * b'(x), and write the individual functions replace 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)

That’s a very long derivative function, but if we organize our thinking in an efficient way, we can quickly and accurately get derivatives of products no matter how long the original function is!

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