# Question Video: Finding the First Derivative of a Trigonometric Function Raised to a Negative Exponent Mathematics • Higher Education

Given that π¦ = (7 cot 5π₯ + 3 csc 6π₯)β»ΒΉ, find dπ¦/dπ₯.

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### Video Transcript

Given that π¦ is equal to seven cot five π₯ plus three cosec six π₯ to the negative one, find dπ¦ by dπ₯.

In this example, we have a reciprocal function. We could write this as a fraction and apply the quotient rule. Alternatively, we could use the chain rule. Letβs look at how we might use the chain rule. This says that if π¦ is a function in π’ and π’ itself is a function in π₯, then the derivative of π¦ with respect to π₯ is dπ¦ by dπ’ times dπ’ by dπ₯. We will let π’ be equal to seven cot of five π₯ plus three cosec six π₯. This means that π¦ is equal to π’ to the negative one. The derivative of π¦ with respect to π’ is fairly straightforward. Itβs negative one times π’ to the negative two. Then we quote the derivative for cot ππ₯ as being negative π cosec squared ππ₯ and the derivative of cosec ππ₯ as being negative π cosec ππ₯ cot ππ₯. So we see dπ’ by dπ₯ is equal to negative 35 cosec squared five π₯ minus 18 cosec six π₯ cot six π₯.

The derivative of π¦ with respect to π₯ is the product of these two. And we recall that we can write negative π’ to the negative two as one over negative π’ squared. We then divide through by negative one and replace π’ with seven cot five π₯ plus three cosec six π₯. And weβve obtained dπ¦ by dπ₯ to be equal to 35 cosec squared five π₯ plus 18 cosec six π₯ cot six π₯ over seven cot five π₯ plus three cosec six π₯ all squared.