Video: Finding the First Derivative of a Trigonometric Function Raised to a Negative Exponent

Given that 𝑦 = (7 cot 5π‘₯ + 3 csc 6π‘₯)⁻¹, find d𝑦/dπ‘₯.

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Video Transcript

Given that 𝑦 is equal to seven cot five π‘₯ plus three cosec six π‘₯ to the negative one, find d𝑦 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. Let’s look at how we might use the chain rule. This says that if 𝑦 is a function in 𝑒 and 𝑒 itself is a function in π‘₯, then the derivative of 𝑦 with respect to π‘₯ is d𝑦 by d𝑒 times d𝑒 by dπ‘₯. We will let 𝑒 be equal to seven cot of five π‘₯ plus three cosec six π‘₯. This means that 𝑦 is equal to 𝑒 to the negative one. The derivative of 𝑦 with respect to 𝑒 is fairly straightforward. It’s negative one times 𝑒 to the negative two. Then we quote the derivative for cot π‘Žπ‘₯ as being negative π‘Ž cosec squared π‘Žπ‘₯ and the derivative of cosec π‘Žπ‘₯ as being negative π‘Ž cosec π‘Žπ‘₯ cot π‘Žπ‘₯. So we see d𝑒 by dπ‘₯ is equal to negative 35 cosec squared five π‘₯ minus 18 cosec six π‘₯ cot six π‘₯.

The derivative of 𝑦 with respect to π‘₯ is the product of these two. And we recall that we can write negative 𝑒 to the negative two as one over negative 𝑒 squared. We then divide through by negative one and replace 𝑒 with seven cot five π‘₯ plus three cosec six π‘₯. And we’ve obtained d𝑦 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.

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