### 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 weβre 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 weβll 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 ππ₯ is π sec squared ππ₯. 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 ππ₯. Itβs π sec ππ₯ times tan ππ₯, 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. 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₯.