### Video Transcript

Find ππ¦ by ππ₯, given that π¦ equals six sin three π₯.

We are looking then to differentiate six sin three π₯ with respect to π₯. And to find this, weβll need to use the fact that the derivative of sin π₯ with respect to π₯ is cos π₯. This is with π₯ 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 π by ππ₯ of six sin π₯ is six cos π₯. But weβre looking for the derivative of six sin three π₯. 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 π§ to be three π₯. Then, as π¦ is equal to six sin three π₯ as weβre told in the question, π¦ is equal to six sin π§. Now, how does this help? Well, the chain rule tells us that the derivative of π¦ with respect to π₯ is the derivative of π¦ with respect to π§ times the derivative of π§ with respect to π₯.

Letβs apply this. We need to find ππ¦ by ππ§. And so we use the expression for π¦ in terms of π§: π¦ equals six sin π§. And just as the derivative with respect to π₯ of six sin π₯ is six cos π₯, the derivative with respect to π§ of six sin π§ is six cos π§.

Now, we just need to find ππ§ by ππ₯. And as π§ equals three π₯, ππ§ by ππ₯ is π by ππ₯ over three π₯, which is three. So altogether, ππ¦ by ππ₯ is six cos π§ times three which is 18 cos π§. And we donβt want ππ¦ by ππ₯ written in terms of some other variable π§. Weβd like it written in terms of π₯ if possible. Using the fact that π§ is three π₯, we see that ππ¦ by ππ₯ is 18 cos three π₯. And this is our final answer.

If we know that the derivative of sin π₯ with respect to π₯ is cos π₯, then we can find the derivative of lots of expressions involving sin π₯ by using the chain rule.