# Question Video: Finding the Second Derivative of Composite Functions Using the Chain Rule Mathematics • Higher Education

Suppose π¦ = π§βΈ and π§ = βπ₯ β 1. Find πΒ²π¦/ππ₯Β².

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### Video Transcript

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

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

And so to find this quantity, we first find ππ¦ by ππ₯. How do we find ππ¦ by ππ₯? 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 ππ¦ by ππ₯ is ππ¦ by ππ§ times ππ§ by ππ₯.

Letβs apply this rule to our problem. π¦ equals π§ to the eight and so ππ¦ by ππ§ is eight π§ to the seven. Now, we need to find ππ§ by ππ₯. 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 ππ¦ by ππ₯ is negative eight π§ to the seven.

Okay, thatβs ππ¦ by ππ₯. But remember weβre looking for π squared π¦ by ππ₯ squared. And weβd already remarked that this is the derivative with respect to π₯ of ππ¦ by ππ₯. Now that weβve found ππ¦ by ππ₯ to be negative eight π§ to the seven, we can see that the quantity weβre looking for π squared π¦ by ππ₯ squared is the derivative π by ππ₯ 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 weβre looking for ππ€ by ππ₯. We happen to have written the chain rule so that ππ¦ by ππ₯ is the subject. But we can change both π¦s, π€s and weβre still expressing the same rule just with different variables. Having written the chain rule in this way, we can just read off that ππ€ by ππ₯ is ππ€ by ππ§ times ππ§ by ππ₯.

Now, letβs clear some room. So what is ππ€ by ππ§? Well, π€ is negative eight π§ to the seven. So ππ€ by ππ§ is negative 56 π§ to the six. And ππ§ by ππ₯ 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 weβre not quite done. Weβd like to have π squared π¦ by ππ₯ 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 ππ₯ 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.