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

Differentiate 𝑓(π‘₯) = 5 sin (5 ln π‘₯).

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Video Transcript

Differentiate 𝑓 of π‘₯ equals five times sin of five times the natural log of π‘₯.

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 𝑒 and 𝑒 is some function in π‘₯, then d𝑦 by dπ‘₯ is equal to d𝑦 by d𝑒 times d𝑒 by dπ‘₯. We’re going to let 𝑒 be equal to five times the natural log of π‘₯. And the derivative of the natural log of π‘₯ is one over π‘₯. So d𝑒 by dπ‘₯ is five times it; it’s five times one over π‘₯, which is simply five over π‘₯. Instead of using 𝑓 of π‘₯, let’s use 𝑦. And this means that 𝑦 is equal to five sin of 𝑒. And the derivative of sin of 𝑒 is cos of 𝑒. So d𝑦 by d𝑒 is five cos 𝑒. And according to the chain rule d𝑦 by dπ‘₯ is the product of these.

Referring back to the original notation, we can say that 𝑓 prime of π‘₯, the derivative, is five over π‘₯ times five cos of 𝑒. We’ll replace 𝑒 with five times the natural log of π‘₯. And we see that, in terms of π‘₯, 𝑓 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 𝑓 prime of π‘₯ is 25 over π‘₯ times the cosine of five times the natural log of π‘₯.

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