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

If 𝑦 = βˆ’9 tan 8π‘₯ sec 8π‘₯, find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 is equal to negative nine tan eight π‘₯ sec eight π‘₯, find d𝑦 by dπ‘₯.

Here, we have a function which is itself the product of two differentiable functions. So we’re going to use the products rule. 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‘₯. So we’ll let 𝑒 be equal to negative nine tan eight π‘₯ and 𝑣 be equal to sec eight π‘₯. We then quote the general result of the derivative of tan π‘Žπ‘₯ is π‘Ž sec squared π‘Žπ‘₯. 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‘₯.

We also quote the general result for the derivative of sec π‘Žπ‘₯. It’s π‘Ž sec π‘Žπ‘₯ times tan π‘Žπ‘₯, which means that d𝑣 by dπ‘₯ is eight sec eight π‘₯ times tan eight π‘₯. We can now substitute everything we know into the formula for the products rule. It’s 𝑒 times d𝑣 by dπ‘₯ plus 𝑣 times d𝑒 by dπ‘₯, which is negative 72 tan squared eight π‘₯ sec eight π‘₯ minus 72 sec cubed eight π‘₯.

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