If 𝑦 is equal to 𝑥 to the power of five sin five 𝑥, determine d𝑦 by d𝑥.
Here, we have a function which is the product of two differentiable functions. We can, therefore, use the product rule to help us evaluate d𝑦 by d𝑥. This says that, for our function 𝑦 is equal to 𝑢 times 𝑣, the derivative of 𝑦 with respect to 𝑥 is equal to 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. We’re going to let 𝑢 be equal to 𝑥 to the power of five and 𝑣 be equal to sin five 𝑥. We begin by evaluating the derivative of 𝑢 with respect to 𝑥. It’s five 𝑥 to the power of four. Similarly, since the derivative of sin 𝑎𝑥 is 𝑎 cos 𝑎𝑥, d𝑣 by d𝑥 is equal to five cos five 𝑥.
Then we substitute into the formula. 𝑢 times d𝑣 by d𝑥 is 𝑥 to the power of five times five cos five 𝑥. And 𝑣 times d𝑢 by d𝑥 is sin five 𝑥 times five 𝑥 to the power of four. And so d𝑦 by d𝑥, in this case, is five 𝑥 to the power of five times cos five 𝑥 plus five 𝑥 to the power of four times sin five 𝑥.