Product rule

In differential calculus, the product rule is a rule that helps calculate derivates that have multiplication.

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 Steps 
Say we have the function .

The two functions being multiplied are  and .

We can set

 and

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The rule needs us to find the derivative of both  and .

We can find  by first using the sum rule to split   into  and . After using the power rule, we have .

To find , we need to find the derivative of , which is , meaning .

Now we can substitute the values into the equation,

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 Proof 
One definition of a derivative is

, and we're trying to find the derivative of , so we can first set  to .

We can't really do much with this so we need to manipulate the equation.

The  part is equal to , meaning it didn't change the value of the equation. Now we can factor,

, and because  approaches ,  is equal to .

, and  and  are just equal to and .

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References

Category:Calculus