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