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

If π¦ = β9 tan 8π₯ sec 8π₯, find dπ¦/dπ₯.

01:12

### Video Transcript

If π¦ is equal to negative nine tan eight π₯ sec eight π₯, find dπ¦ by dπ₯.

Here, we have a function which is itself the product of two differentiable functions. So weβre going to use the products rule. 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₯. So weβll let π’ be equal to negative nine tan eight π₯ and π£ be equal to sec eight π₯. We then quote the general result of the derivative of tan ππ₯ is π sec squared ππ₯. 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π₯. Itβs π sec ππ₯ times tan ππ₯, which means that dπ£ 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. Itβs 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₯.