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

Differentiate π(π₯) = 5 sin (5 ln π₯).

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### Video Transcript

Differentiate π of π₯ equals five times sin of five times the natural log of π₯.

This is a composite function. Itβs a function of a function. And we can, therefore, use the chain rule to find its derivative. This says that if π¦ is some function in π’ and π’ is some function in π₯, then dπ¦ by dπ₯ is equal to dπ¦ by dπ’ times dπ’ by dπ₯. Weβre going to let π’ be equal to five times the natural log of π₯. And the derivative of the natural log of π₯ is one over π₯. So dπ’ by dπ₯ is five times it; itβs five times one over π₯, which is simply five over π₯. Instead of using π of π₯, letβs use π¦. And this means that π¦ is equal to five sin of π’. And the derivative of sin of π’ is cos of π’. So dπ¦ by dπ’ is five cos π’. And according to the chain rule dπ¦ by dπ₯ is the product of these.

Referring back to the original notation, we can say that π prime of π₯, the derivative, is five over π₯ times five cos of π’. Weβll replace π’ with five times the natural log of π₯. And we see that, in terms of π₯, π prime of π₯ is five over π₯ times five cos of five times the natural log of π₯. And then we simplify and we see theat π prime of π₯ is 25 over π₯ times the cosine of five times the natural log of π₯.