Question Video: Differentiating Functions Involving Trigonometric Ratios and Natural Logarithms Using the Chain Rule Mathematics

Video Transcript

Differentiate 𝑓 of 𝑥 equals five times sin of five times the natural log of 𝑥.

This is a composite function. It’s a function of a function. And we can, therefore, use the chain rule to find its derivative. This says that if 𝑦 is some function in 𝑢 and 𝑢 is some function in 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. We’re going to let 𝑢 be equal to five times the natural log of 𝑥. And the derivative of the natural log of 𝑥 is one over 𝑥. So d𝑢 by d𝑥 is five times it; it’s five times one over 𝑥, which is simply five over 𝑥. Instead of using 𝑓 of 𝑥, let’s use 𝑦. And this means that 𝑦 is equal to five sin of 𝑢. And the derivative of sin of 𝑢 is cos of 𝑢. So d𝑦 by d𝑢 is five cos 𝑢. And according to the chain rule d𝑦 by d𝑥 is the product of these.

Referring back to the original notation, we can say that 𝑓 prime of 𝑥, the derivative, is five over 𝑥 times five cos of 𝑢. We’ll replace 𝑢 with five times the natural log of 𝑥. And we see that, in terms of 𝑥, 𝑓 prime of 𝑥 is five over 𝑥 times five cos of five times the natural log of 𝑥. And then we simplify and we see theat 𝑓 prime of 𝑥 is 25 over 𝑥 times the cosine of five times the natural log of 𝑥.