If 𝑦 is equal to negative nine tan eight 𝑥 sec eight 𝑥, find d𝑦 by d𝑥.
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 𝑢 and 𝑣 is 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. So we’ll let 𝑢 be equal to negative nine tan eight 𝑥 and 𝑣 be equal to sec eight 𝑥. We then quote the general result of the derivative of tan 𝑎𝑥 is 𝑎 sec squared 𝑎𝑥. And this means the derivative of negative nine tan eight 𝑥 is negative nine times eight sec squared eight 𝑥, which is negative 72 sec squared eight 𝑥.
We also quote the general result for the derivative of sec 𝑎𝑥. It’s 𝑎 sec 𝑎𝑥 times tan 𝑎𝑥, which means that d𝑣 by d𝑥 is eight sec eight 𝑥 times tan eight 𝑥. We can now substitute everything we know into the formula for the products rule. It’s 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥, which is negative 72 tan squared eight 𝑥 sec eight 𝑥 minus 72 sec cubed eight 𝑥.