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

Find dπ¦/dπ₯ if π¦ = tan (5 cot π₯).

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### Video Transcript

Find dπ¦ by dπ₯ if π¦ 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 dπ¦ by dπ₯ is equal to dπ¦ by dπ’ multiplied by dπ’ dπ₯. We begin by letting π’ equal five cot π₯. We know that the derivative of cot π₯ is negative cosec squared π₯. This means that dπ’ by dπ₯ is equal to negative five cosec squared π₯. Replacing five cot π₯ with π’ in our initial equation gives us π¦ is equal to tan π’.

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