If 𝑦 equals cos cubed of five minus four 𝑥, find d𝑦 by d𝑥.
Let’s begin by rewriting our function slightly. We’re going to write it as cos of five minus four 𝑥 all cubed. And when we write it like this, we notice we have a composite function. That’s a function of another function. And so to find d𝑦 by d𝑥, that’s the derivative of 𝑦 with respect to 𝑥, we need to use the chain rule. This says that if 𝑦 is some function of 𝑢 and 𝑢 itself is some function of 𝑥, then the derivative of 𝑦 with respect to 𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. We let 𝑢 be equal to the inner part of our composite function. So 𝑢 is equal to cos of five minus four 𝑥. And that means that 𝑦 is equal to 𝑢 cubed. We can see from the chain rule that we need to evaluate d𝑦 by d𝑢 and d𝑢 by d𝑥.
Well, d𝑦 by d𝑢 is quite straightforward. We multiply that entire term by the exponent and then reduce that exponent by one. So d𝑦 by d𝑢 is three 𝑢 squared. But what about the derivative of cos of five minus four 𝑥? Well, we’re going to use the chain rule to evaluate this too. We’ll let 𝑣 be equal to five minus four 𝑥, such that 𝑢 is equal to cos of 𝑣. We can then reword our chain rule slightly. And we say that the derivative of 𝑢 with respect to 𝑥 here will be equal to the derivative of 𝑢 with respect to 𝑣 times the derivative of 𝑣 with respect to 𝑥. Then we see that d𝑣 by d𝑥 is negative four. And of course, the derivative of cos of 𝑣 with respect to 𝑣 is negative sin 𝑣. d𝑢 by d𝑥 is then the product of these. It’s negative four multiplied by negative sin 𝑣. That’s four sin 𝑣.
And since 𝑣 is equal to five minus four 𝑥 and we want d𝑢 by d𝑥 to be in terms of 𝑥, we can say that this is equal to four sin of five minus four 𝑥. Now that we know d𝑢 by d𝑥 and d𝑦 by d𝑢, we can work out d𝑦 by d𝑥 by finding their product. That’s three 𝑢 squared times four sin of five minus four 𝑥. We can multiply the three and the four to give us a coefficient of 12. And then we replace 𝑢 with our earlier substitution with cos of five minus four 𝑥. And so d𝑦 by d𝑥 is 12 times cos of five minus four 𝑥 all squared times sin of five minus four 𝑥. And then we know that cos of five minus four 𝑥 squared can be written as cos squared of five minus four 𝑥. Writing our terms in increasing powers, and we see that d𝑦 by d𝑥 is 12 sin of five minus four 𝑥 times cos squared of five minus four 𝑥.