Find 𝑑𝑦 by 𝑑𝑥, given that 𝑦 equals six sin three 𝑥.
We are looking then to differentiate six sin three 𝑥 with respect to 𝑥. And to find this, we’ll need to use the fact that the derivative of sin 𝑥 with respect to 𝑥 is cos 𝑥. This is with 𝑥 in radians. Using the fact that the derivative of a number times a function is that number times the derivative of the function, we can see that 𝑑 by 𝑑𝑥 of six sin 𝑥 is six cos 𝑥. But we’re looking for the derivative of six sin three 𝑥. How do we find this? Well, we have to use the chain rule.
To make it easier to apply the chain rule, we will define a new variable 𝑧 to be three 𝑥. Then, as 𝑦 is equal to six sin three 𝑥 as we’re told in the question, 𝑦 is equal to six sin 𝑧. Now, how does this help? Well, the chain rule tells us that the derivative of 𝑦 with respect to 𝑥 is the derivative of 𝑦 with respect to 𝑧 times the derivative of 𝑧 with respect to 𝑥.
Let’s apply this. We need to find 𝑑𝑦 by 𝑑𝑧. And so we use the expression for 𝑦 in terms of 𝑧: 𝑦 equals six sin 𝑧. And just as the derivative with respect to 𝑥 of six sin 𝑥 is six cos 𝑥, the derivative with respect to 𝑧 of six sin 𝑧 is six cos 𝑧.
Now, we just need to find 𝑑𝑧 by 𝑑𝑥. And as 𝑧 equals three 𝑥, 𝑑𝑧 by 𝑑𝑥 is 𝑑 by 𝑑𝑥 over three 𝑥, which is three. So altogether, 𝑑𝑦 by 𝑑𝑥 is six cos 𝑧 times three which is 18 cos 𝑧. And we don’t want 𝑑𝑦 by 𝑑𝑥 written in terms of some other variable 𝑧. We’d like it written in terms of 𝑥 if possible. Using the fact that 𝑧 is three 𝑥, we see that 𝑑𝑦 by 𝑑𝑥 is 18 cos three 𝑥. And this is our final answer.
If we know that the derivative of sin 𝑥 with respect to 𝑥 is cos 𝑥, then we can find the derivative of lots of expressions involving sin 𝑥 by using the chain rule.