Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Chain Rule Mathematics • Higher Education

Find d𝑦/dπ‘₯ if 𝑦 = tan (5 cot π‘₯).

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Video Transcript

Find d𝑦 by dπ‘₯ if 𝑦 is equal to the tan of five cot π‘₯.

Our function here is a composite function, so we will use the chain rule to differentiate it. This states that d𝑦 by dπ‘₯ is equal to d𝑦 by d𝑒 multiplied by d𝑒 dπ‘₯. We begin by letting 𝑒 equal five cot π‘₯. We know that the derivative of cot π‘₯ is negative cosec squared π‘₯. This means that d𝑒 by dπ‘₯ is equal to negative five cosec squared π‘₯. Replacing five cot π‘₯ with 𝑒 in our initial equation gives us 𝑦 is equal to tan 𝑒.

Once again, we know that the derivative of tan π‘₯ is sec squared π‘₯. If 𝑦 is equal to tan 𝑒, then d𝑦 by d𝑒 is equal to sec squared 𝑒. We now have expressions for d𝑦 by d𝑒 and d𝑒 by dπ‘₯. In order to work out d𝑦 by dπ‘₯, both of these need to be in terms of π‘₯. We can, therefore, rewrite d𝑦 by d𝑒 as sec squared of five cot π‘₯. d𝑦 by dπ‘₯ is, therefore, equal to negative five cosec squared π‘₯ multiplied by sec squared of five cot π‘₯.

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