If 𝑦 equals sin of 17 over 𝑥 to the ninth power, find d𝑦 by d𝑥.
What we have here is an example of a composite function. We have the function 17 over 𝑥 to the ninth power, and then we’re taking the sine of this function. So, we have a function of a function. In order to find the derivative of a composite function, we need to recall the chain rule. The chain rule says that if 𝑦 is some function of 𝑢 and 𝑢 is some function of 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢 multiplied by d𝑢 by d𝑥. Let’s see what applying the chain would look like in this question.
We’re going to let 𝑢 equal the inner part of our composite function. So, that’s 17 over 𝑥 to the ninth power. Or we can write this using negative exponents as 17𝑥 to the power of negative nine. Then, as 𝑦 is equal to sin of 17 over 𝑥 to the ninth power and 𝑢 is equal to 17 over 𝑥 to the ninth power, 𝑦 will be equal to sin of 𝑢. And so, we have 𝑦 as a function of 𝑢 and 𝑢 as a function of 𝑥. In order to apply the chain rule, we need to find their derivatives. That’s d𝑢 by d𝑥 and d𝑦 by d𝑢.
To find d𝑢 by d𝑥, we need to recall the power rule of differentiation, which tells us that to differentiate a power of 𝑥, we multiply by that power or exponent, and then reduce the exponent by one. So, the derivative of 17𝑥 to the power of negative nine will be 17 multiplied by negative nine 𝑥 to the power of negative 10. Which simplifies to negative 153𝑥 to the power of negative 10. To find d𝑦 by d𝑢, we need to recall our standard results for differentiating trigonometric functions. And remember here, the argument of the trigonometric function must be given in radians.
We recall that the derivative of sin 𝑢 with respect to 𝑢 is cos 𝑢. So, we have that d𝑦 by d𝑢 equals cos 𝑢. Now, we can substitute into the chain rule. This tells us that d𝑦 by d𝑥 is equal to d𝑦 by d𝑢, so that’s cos 𝑢, multiplied by d𝑢 by d𝑥, which is negative 153𝑥 to the power of negative 10. Now, we have an expression for d𝑦 by d𝑥, but it’s in terms of two variables, the variable 𝑢 and the variable 𝑥. And we need it to be in terms of our original variable only. So, the final step is to reverse our substitution.
Remember that we defined 𝑢 to be equal to 17 over 𝑥 to the ninth power. So, we undo our substitution by replacing 𝑢 with 17 over 𝑥 to the ninth power. And at the same time, we can write negative 153𝑥 to the power of negative 10 as negative 153 over 𝑥 to the 10th power. And so, we have our expression for d𝑦 by d𝑥. By using the chain rule which, remember, tells us how to find the derivative of a composite function, we’ve found that d𝑦 by d𝑥 is equal to negative 153 over 𝑥 to the 10th power cos of 17 over 𝑥 to the ninth power.