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

When you first start learning calculus concepts, you start with how to take the derivatives of various functions. You will learn that the derivative of sin(x) is cos(x), that the derivative of ax^n is anx^(n-1), and several other basic function rules that you saw through algebra and trigonometry. Once you’re familiar with the derivatives of individual functions, you’re looking at the derivatives of those functions, which drastically expands the range of functions you can take derivatives from.

However, going from taking derivatives of basic functions to taking derivatives of functions is a big step in complexity. Because of this big step, how complicated the process is, many students feel overwhelmed and have a lot of trouble understanding the material. Unfortunately, many instructors don’t give students methods to solve these problems, but we do! Let’s begin.

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 would mean 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) looks like this:

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

We call this formula the product rule. It’s more complicated than any previous derivative formula you’ve seen in your calculus sequence so far. However, if you write out all the functions you’re dealing with BEFORE you try to write out f'(x), your speed and accuracy will improve greatly. So the first step is to write out what a(x) is and what b(x) is. Then find the derivatives a'(x) and b'(x) next to it. Once you’ve written it all out, there’s nothing left to think about and just fill in the blanks in the product rule formula. That is all.

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

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 and b(x) = 3cos(x) – 2x² + 4x + 5, since these two functions are multiplied together to form f(x). Then we simply write on the side of our paper:

a(x) = 5sin(x) + 4x³ – 16x

b(x) = 3cos(x) – 2x² + 4x + 5

Once written out separately from each other, we now find the derivative of a(x) and b(x) separately just below it. 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 out in an organized way, we don’t have to remember anything! All work related to this issue has been completed. We just need to write the four functions in the correct order that the product rule gives us.

Finally, write out the basic form of the product rule f'(x) = a'(x) * b(x) + a(x) * b'(x) and write the corresponding functions in place of a. (x), a'(x), b(x) and b'(x). Back to where we are dealing with 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 derivative function, but if we organize our thinking efficiently, we can quickly and accurately take derivatives of products, regardless of the length of the original function!

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