Video: Differentiating Functions Involving Reciprocal Trigonometric Ratios Using the Product Rule

Given 𝑦 = (π‘₯ + 3)(9π‘₯ + csc π‘₯), find d𝑦/dπ‘₯.

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Video Transcript

Given 𝑦 is equal to π‘₯ plus three times nine π‘₯ plus cosec π‘₯, find d𝑦 by dπ‘₯.

Here, we have an expression which is the product of two functions. We’re therefore going to use the product rule to calculate d𝑦 by dπ‘₯. This says that the derivative of the product of two differentiable functions 𝑒 and 𝑣 is 𝑒 times d𝑣 by dπ‘₯ plus 𝑣 times d𝑒 by dπ‘₯. We therefore let 𝑒 be equal to π‘₯ plus three and 𝑣 be equal to nine π‘₯ plus cosec π‘₯. The derivative of π‘₯ plus three is simply one. But what about d𝑣 by dπ‘₯? Well, we know that the derivative of nine π‘₯ is nine. And the derivative of cosec π‘₯ is negative cosec π‘₯ cot π‘₯. So d𝑣 by dπ‘₯ is equal to nine minus cosec π‘₯ cot π‘₯. Let’s substitute what we have into the formula for the product rule. We see that d𝑦 by dπ‘₯ is equal to π‘₯ plus three times nine minus cosec π‘₯ cot π‘₯ plus nine π‘₯ plus cosec π‘₯ times one. We distribute our parentheses and then collect like terms. And we see that d𝑦 by dπ‘₯ is 18π‘₯ minus π‘₯ plus three times cosec π‘₯ cot π‘₯ plus cosec π‘₯ plus 27.

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