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

Suppose 𝑦 = 𝑧⁸ and 𝑧 = βˆ’π‘₯ βˆ’ 1. Find 𝑑²𝑦/𝑑π‘₯Β².

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Video Transcript

Suppose 𝑦 equals 𝑧 to the eight and 𝑧 equals minus π‘₯ minus one. Find 𝑑 squared 𝑦 by 𝑑π‘₯ squared.

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

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

Let’s apply this rule to our problem. 𝑦 equals 𝑧 to the eight and so 𝑑𝑦 by 𝑑𝑧 is eight 𝑧 to the seven. Now, we need to find 𝑑𝑧 by 𝑑π‘₯. And this is straightforward because we know that 𝑧 equals minus π‘₯ minus one. The derivative of minus π‘₯ minus one with respect to π‘₯ is negative one. So 𝑑𝑦 by 𝑑π‘₯ is negative eight 𝑧 to the seven.

Okay, that’s 𝑑𝑦 by 𝑑π‘₯. But remember we’re looking for 𝑑 squared 𝑦 by 𝑑π‘₯ squared. And we’d already remarked that this is the derivative with respect to π‘₯ of 𝑑𝑦 by 𝑑π‘₯. Now that we’ve found 𝑑𝑦 by 𝑑π‘₯ to be negative eight 𝑧 to the seven, we can see that the quantity we’re looking for 𝑑 squared 𝑦 by 𝑑π‘₯ squared is the derivative 𝑑 by 𝑑π‘₯ of negative eight 𝑧 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 𝑀 to be equal to negative eight 𝑧 to the seven. Then we’re looking for 𝑑𝑀 by 𝑑π‘₯. We happen to have written the chain rule so that 𝑑𝑦 by 𝑑π‘₯ is the subject. But we can change both 𝑦s, 𝑀s and we’re still expressing the same rule just with different variables. Having written the chain rule in this way, we can just read off that 𝑑𝑀 by 𝑑π‘₯ is 𝑑𝑀 by 𝑑𝑧 times 𝑑𝑧 by 𝑑π‘₯.

Now, let’s clear some room. So what is 𝑑𝑀 by 𝑑𝑧? Well, 𝑀 is negative eight 𝑧 to the seven. So 𝑑𝑀 by 𝑑𝑧 is negative 56 𝑧 to the six. And 𝑑𝑧 by 𝑑π‘₯ we found earlier. Using the fact that 𝑧 equals minus π‘₯ minus one, we found it to be negative one. Simplifying then, we get 56 𝑧 to the six.

But we’re not quite done. We’d like to have 𝑑 squared 𝑦 by 𝑑π‘₯ squared written in terms of π‘₯ rather than the auxiliary variable 𝑧. And we can do this because we have an expression for 𝑧 in terms of π‘₯ minus π‘₯ minus one. And we can substitute this expression for 𝑧.

Our final answer is therefore that 𝑑 squared 𝑦 by 𝑑π‘₯ 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.

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