### Video Transcript

Suppose π¦ equals π§ to the eight and π§ equals minus π₯ minus one. Find π squared π¦ by ππ₯ squared.

This π squared π¦ by ππ₯ squared is the second derivative of π¦ with respect to π₯. If you differentiate π¦ with respect to π₯, you get ππ¦ by ππ₯. And if you differentiate again with respect to π₯, you get this: π squared π¦ by ππ₯ squared. In other words, this is the derivative with respect to π₯ of ππ¦ by ππ₯.

And so to find this quantity, we first find ππ¦ by ππ₯. How do we find ππ¦ by ππ₯? Well, we have π¦ in terms of another variable π§ and π§ in terms of π₯. So it looks like a job for the chain rule. The chain rule tells us that ππ¦ by ππ₯ is ππ¦ by ππ§ times ππ§ by ππ₯.

Letβs apply this rule to our problem. π¦ equals π§ to the eight and so ππ¦ by ππ§ is eight π§ to the seven. Now, we need to find ππ§ by ππ₯. And this is straightforward because we know that π§ equals minus π₯ minus one. The derivative of minus π₯ minus one with respect to π₯ is negative one. So ππ¦ by ππ₯ is negative eight π§ to the seven.

Okay, thatβs ππ¦ by ππ₯. But remember weβre looking for π squared π¦ by ππ₯ squared. And weβd already remarked that this is the derivative with respect to π₯ of ππ¦ by ππ₯. Now that weβve found ππ¦ by ππ₯ to be negative eight π§ to the seven, we can see that the quantity weβre looking for π squared π¦ by ππ₯ squared is the derivative π by ππ₯ of negative eight π§ to the seven.

And the easiest way to find this derivative is again using the chain rule. The easiest way to see this is probably to define another variable π€ to be equal to negative eight π§ to the seven. Then weβre looking for ππ€ by ππ₯. We happen to have written the chain rule so that ππ¦ by ππ₯ is the subject. But we can change both π¦s, π€s and weβre still expressing the same rule just with different variables. Having written the chain rule in this way, we can just read off that ππ€ by ππ₯ is ππ€ by ππ§ times ππ§ by ππ₯.

Now, letβs clear some room. So what is ππ€ by ππ§? Well, π€ is negative eight π§ to the seven. So ππ€ by ππ§ is negative 56 π§ to the six. And ππ§ by ππ₯ we found earlier. Using the fact that π§ equals minus π₯ minus one, we found it to be negative one. Simplifying then, we get 56 π§ to the six.

But weβre not quite done. Weβd like to have π squared π¦ by ππ₯ squared written in terms of π₯ rather than the auxiliary variable π§. And we can do this because we have an expression for π§ in terms of π₯ minus π₯ minus one. And we can substitute this expression for π§.

Our final answer is therefore that π squared π¦ by ππ₯ squared is 56 minus π₯ minus one to the six. We could also have factored out negative one inside the parentheses to get an answer of 56π₯ plus one to the six. This answer would also be correct.