WEBVTT
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If π¦ is equal to negative nine tan eight π₯ sec eight π₯, find dπ¦ by dπ₯.
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Here, we have a function which is itself the product of two differentiable functions.
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So weβre going to use the products rule.
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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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So weβll let π’ be equal to negative nine tan eight π₯ and π£ be equal to sec eight π₯.
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We then quote the general result of the derivative of tan ππ₯ is π sec squared ππ₯.
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And this means the derivative of negative nine tan eight π₯ is negative nine times eight sec squared eight π₯, which is negative 72 sec squared eight π₯.
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We also quote the general result for the derivative of sec ππ₯.
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Itβs π sec ππ₯ times tan ππ₯, which means that dπ£ by dπ₯ is eight sec eight π₯ times tan eight π₯.
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We can now substitute everything we know into the formula for the products rule.
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Itβs π’ times dπ£ by dπ₯ plus π£ times dπ’ by dπ₯, which is negative 72 tan squared eight π₯ sec eight π₯ minus 72 sec cubed eight π₯.