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Product Rule

Product Rule

The product rule is one of the differentiation rules you need to know. This rule is used when you are differentiating the products of two functions.

Product rule formula and function notation

There is a formula that you can follow to use the product rule. It is important to try and remember this formula as it is not usually in exam formula booklets. If when u and v are functions of x then the product rule formula is:

This can also be written in function notation,

If then

Examples of the product rule

To understand the product rule further, let's look through some examples of using it.

If find :

First, you can start by looking at your product rule formula and finding each aspect of the formula:

If , and

To find and you can differentiate your u and v:

Now that you have all aspects of your formula, you can solve it to find :

You may also be asked to differentiate a trigonometric function using the product rule.

If find


As before, you can start by finding each part of the formula:

If , and

Now you can differentiate your u and v to find and :

Finally, you can substitute each part into your formula to find :

You may even be asked to use the product rule in function notation, so let's work through an example.

If find

Once again, you can start by breaking down the formula for the product rule in function notation and finding each part.

If , and

Next, you can differentiate and to find the derivatives, and :

Now that you have each part of the formula, you can solve it to find :

If find

To begin with, let's look at the product rule formula and find each aspect of it:

If , and

Let's differentiate the terms to find and :

Now you can put each part into the formula to find :

Product Rule - Key takeaways

  • The product rule is one of the differentiation rules.

  • The product rule can be used when differentiating the products of two functions.

  • When using the product rule, you can either use the formula in y form or in the function notation form.

  • You may also need to differentiate trigonometric functions using the product rule.

Frequently Asked Questions about Product Rule

The product rule is one of the differentiation rules that is used when you are differentiating the products of two functions.

You use the product rule when you are dealing with the products of two functions, and you use the chain rule when you are differentiating a composite function, also known as a function of a function.

The product rule can be derived from the definition of a derivative,  dy/dx f(x) = lim (f(x+h)-f(x))/h

Final Product Rule Quiz

Question

What is the product rule?

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Answer

The product rule is a rule used for differentiation.

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Question

When do you use the product rule?

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Answer

You can use the product rule when you are differentiating the products of two functions.

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