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.
