# Question Video: Using the Product Rule Mathematics • Higher Education

The product rule says that (ππ)β² = πβ²π + ππβ². Use this to derive a formula for the derivative (ππ β).

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### Video Transcript

The product rule says that the derivative of ππ is equal to the derivative of π times π plus π times the derivative of π. Use this to derive a formula for the derivative of π times π times β.

In this question, weβve been given the product rule and asked to use it to find a formula for the derivative of the product of three functions. These are π, π, and β. Weβre going to begin by splitting π times π times β up. Weβre going to write it as ππ times β. Remember since multiplication is commutative, we could have alternatively written it as π times πβ and we will get the same answer either way. So we can say that the derivative of ππβ is equal to the derivative of ππ times β.

And weβre now going to apply the product rule. We can see that this is equal to the derivative of ππ times β plus ππ times the derivative of β. And now, we spot that the first term we have is the derivative of ππ. We know though by the definition of the product rule that this is the same as the derivative of π times π plus π times the derivative of π. So we replace this in our formula. And weβre going to distribute these parentheses.

When we do, we see that the formula for the derivative of ππβ is the derivative of π times π times β plus π times the derivative of π times β plus π times π times the derivative of β. You might also like to see if you can apply this idea to help you find a formula for the derivative of the product of four functions, say ππβπ.