Understand the Derivative Product Rule

As you may know, the operation that apply to numbers do not necessary apply for derivatives too. A good example is the product operation. The derivative product rule is a little bit more special and hard to understand and prove.

The product rule for derivatives is used when we have to differentiate two functions that are being multiplied together.

Basically, the product rule derivatives are suffering the following operation: 
product rule

In words, the product rule definition says that: the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Let's see some Product Rule Examples

In order to get used with the rule, we provide you a series of product rule examples and product rule problems.

1.) Find f'(x): product rule examples

Solution: product rule example

The first term is: product rule problems

The second term is: product rule problem

Then, we add them together and obtain our derivative: obtain derivative

2.) Differentiate: example of product rule

Solution: examples of product rule

3.) The third example constitute in the zero product rule: any derivative multiplied with zero results zero.

4.) In the fourth example we will present the triple product rule. Basically, it is the product rule with 3 terms.
triple product rule

After we saw the product rule with three terms, it's time we have a look at the product rule proof.

Proof Of Product Rule

The proof of product rule is not so easy to make, but if you pay attention on the demonstration you will surely understand it.

The definition of derivative:
definition of derivative

The key is to subtract and add a term: examples and formula of product rule
solve with product rule

By definition, product rule examples
and product rule proof and examples

Also how to use product rule
and product rule for derivative

In conclusion, we reach the product rule formulaproduct rule formula

Now that the proof of the product rule is complete, it's time we find out a few more facts about our subject.

For radicals, the product rule derivative is suffering the same transformations, so the product rule for radicals is basically the same.

The equality in the formula is true in both sides, which means that the reverse product rule is also valid.


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