Question Video: Using The Product Rule | Nagwa

Video Transcript

The product rule says that the derivative of 𝑓𝑔 is equal to the derivative of 𝑓 times 𝑔 plus 𝑓 times the derivative of 𝑔. Use this to derive a formula for the derivative of 𝑓 times 𝑔 times ℎ.

In this question, we’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 𝑓, 𝑔, and ℎ. We’re going to begin by splitting 𝑓 times 𝑔 times ℎ up. We’re going to write it as 𝑓𝑔 times ℎ. Remember since multiplication is commutative, we could have alternatively written it as 𝑓 times 𝑔ℎ and we will get the same answer either way. So we can say that the derivative of 𝑓𝑔ℎ is equal to the derivative of 𝑓𝑔 times ℎ.

And we’re now going to apply the product rule. We can see that this is equal to the derivative of 𝑓𝑔 times ℎ plus 𝑓𝑔 times the derivative of ℎ. And now, we spot that the first term we have is the derivative of 𝑓𝑔. We know though by the definition of the product rule that this is the same as the derivative of 𝑓 times 𝑔 plus 𝑓 times the derivative of 𝑔. So we replace this in our formula. And we’re going to distribute these parentheses.

When we do, we see that the formula for the derivative of 𝑓𝑔ℎ is the derivative of 𝑓 times 𝑔 times ℎ plus 𝑓 times the derivative of 𝑔 times ℎ plus 𝑓 times 𝑔 times the derivative of ℎ. 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 𝑓𝑔ℎ𝑖.