Question Video: Finding the First Derivative of a Composite Function Using the Chain Rule at a Point Mathematics • Higher Education

Given that 𝑦 = −6𝑧² − 23 and 𝑧 = −(4/𝑥), determine 𝑑𝑦/𝑑𝑥.

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Video Transcript

Given that 𝑦 equals negative six 𝑧 squared minus 23 and 𝑧 is equal to negative four of 𝑥, determine 𝑑𝑦 by 𝑑𝑥.

We’re looking for the derivative of 𝑦 with respect to 𝑥 and we’re given 𝑦 in terms of another variable 𝑧 and 𝑧 in terms of 𝑥. So this looks like a job for the chain rule, which says that the derivative of 𝑦 with respect to 𝑥 is equal to the derivative of 𝑦 with respect to some other variable 𝑧 times the derivative of 𝑧 with respect to 𝑥.

Let’s apply the chain rule to our problem. We need to find 𝑑𝑦 by 𝑑𝑧. That is the derivative of 𝑦 with respect to 𝑧. Luckily, we have 𝑦 written in terms of 𝑧. So this is straightforward. 𝑦 equals negative six 𝑧 squared minus 23. We differentiate both sides with respect to 𝑧. Using the fact that the derivative of a difference of functions is the difference of their derivatives and using the formula for the derivative of a number times a power with respect to the base of that power and also the fact that the derivative of a constant function is zero, we get negative 12𝑧.

Having found 𝑑𝑦 by 𝑑𝑧, we now move on to finding 𝑑𝑧 by 𝑑𝑥. And to find 𝑑𝑧 by 𝑑𝑥, we use the relation between 𝑧 and 𝑥. 𝑧 equals negative four over 𝑥 and we can write this in exponent notation as 𝑧 equals negative four 𝑥 to the negative one. We can now apply our rule about the derivative of a number times a power of a variable with respect to that variable.

To make things clearer, we’ll change the 𝑧 in our rule to 𝑥. It’s important to note that we could replace 𝑧 by any letter that we would like and it would still express the same rule. But we choose to replace 𝑧 by 𝑥 because we’re going to be differentiating with respect to 𝑥. Applying our rule, we find 𝑑𝑧 by 𝑑𝑥 to be four 𝑥 to the negative two or if you want to avoid negative exponents four over 𝑥 squared.

Using this expression for 𝑑𝑧 by 𝑑𝑥, we see that 𝑑𝑦 by 𝑑𝑥 is negative 12 times 𝑧 times four over 𝑥 squared. Now, it’s just a case of simplifying this expression. And as part of the simplification, we realize that we have 𝑑𝑦 by 𝑑𝑥 written not only in terms of 𝑥, but also in terms of the variable 𝑧. And we’d like it to be solely in terms of 𝑥 if that’s possible. And it is possible because 𝑧 is negative four over 𝑥.

Substituting negative four over 𝑥 for 𝑧, we get an expression for 𝑑𝑦 by 𝑑𝑥 in terms of 𝑥 alone. And we can simplify this expression with negative 12 times negative four times four making 192 in the numerator and 𝑥 times 𝑥 squared making 𝑥 cubed in the denominator.

𝑑𝑦 by 𝑑𝑥 is then 192 over 𝑥 cubed.