Video: Finding the First Derivative of Polynomial Functions Using the Chain Rule

Video Transcript

Find the derivative of the function 𝑦 equals five 𝑥 squared minus six all to the power of six.

One thing we could do is to take this five 𝑥 squared minus six all to the power of six and expand it binomially — that is using Pascal’s triangle. Having done this, we would get a polynomial in 𝑥, which we could then differentiate term by term in the normal way. But this will take a long time both to binomially expand the expression and to differentiate all the terms you get.

Surely, there’s a better way. Luckily, there is a better way. Let 𝑧 equal five 𝑥 squared minus six — that is the expression inside the parenthesis. Then, simply, by a substituting 𝑧 for five 𝑥 squared minus six, we see that 𝑦 is equal to 𝑧 to the sixth.

How does this help us find the derivative of the function with respect to 𝑥 though? Well, we can use the chain rule, which says that the derivative of 𝑦 with respect to 𝑥 is the derivative of 𝑦 with respect to some other variable 𝑧 times the derivative of 𝑧 with respect to 𝑥. It’s almost like the derivatives 𝑑𝑦 by 𝑑𝑧 and 𝑑𝑧 by 𝑑𝑥 are fractions and the 𝑑𝑧s cancel.

Applying the chain rule, we now need to find 𝑑𝑦 by 𝑑𝑧 and 𝑑𝑧 by 𝑑𝑥 and multiply them together. Let’s start with 𝑑𝑦 by 𝑑𝑧. 𝑦 is equal to 𝑧 to the sixth. And so differentiating 𝑦 with respect to 𝑧, we get six times 𝑧 to the five. Here, we’ve used the fact that just like the derivative of a power of 𝑥 with respect to 𝑥, you find the derivative of a power of 𝑧 with respect to 𝑧 by taking a copy of the exponents down to the front and multiplying by it and then reducing the exponent by one.

It doesn’t matter what the variable is — whether it’s 𝑥 or 𝑧 or even capital 𝑄. Having found 𝑑𝑦 by 𝑑𝑧, we now need to find 𝑑𝑧 by 𝑑𝑥. Well, we have that 𝑧 is equal to five 𝑥 squared minus six. And differentiating this, we get that 𝑑𝑧 by 𝑑𝑥 is 10𝑥. Hence, we see that 𝑑𝑦 by 𝑑𝑥 is seven [six] 𝑧 to the five times 10𝑥, which is 60𝑥𝑧 to the five. We’re almost done here. The last thing to say is that we want 𝑑𝑦 by 𝑑𝑥 in terms of 𝑥 alone. And we can do this by substituting our expression for 𝑧 in terms of 𝑥. 𝑧 is five 𝑥 squared minus six. And so 60𝑥𝑧 to the five is 60𝑥 five 𝑥 squared minus six to the five.

And this is our final answer. Well, as we said at the beginning of the video, we can solve this problem without using the chain rule. Using the chain rule is much more efficient. For many differentiation problems, there’s no way of avoiding the chain rule. So it’s really worth remembering.