WEBVTT
00:00:00.520 --> 00:00:05.910
Find ππ¦ by ππ₯, given that π¦ equals six sin three π₯.
00:00:07.150 --> 00:00:11.580
We are looking then to differentiate six sin three π₯ with respect to π₯.
00:00:12.780 --> 00:00:18.520
And to find this, weβll need to use the fact that the derivative of sin π₯ with respect to π₯ is cos π₯.
00:00:19.580 --> 00:00:21.660
This is with π₯ in radians.
00:00:22.630 --> 00:00:33.520
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₯.
00:00:34.470 --> 00:00:38.320
But weβre looking for the derivative of six sin three π₯.
00:00:39.440 --> 00:00:40.630
How do we find this?
00:00:40.880 --> 00:00:42.410
Well, we have to use the chain rule.
00:00:43.550 --> 00:00:48.350
To make it easier to apply the chain rule, we will define a new variable π§ to be three π₯.
00:00:49.280 --> 00:00:56.410
Then, as π¦ is equal to six sin three π₯ as weβre told in the question, π¦ is equal to six sin π§.
00:00:57.500 --> 00:00:58.810
Now, how does this help?
00:00:59.560 --> 00:01:08.800
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₯.
00:01:09.970 --> 00:01:11.250
Letβs apply this.
00:01:11.910 --> 00:01:14.010
We need to find ππ¦ by ππ§.
00:01:14.990 --> 00:01:19.810
And so we use the expression for π¦ in terms of π§: π¦ equals six sin π§.
00:01:20.800 --> 00:01:31.100
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§.
00:01:33.550 --> 00:01:36.020
Now, we just need to find ππ§ by ππ₯.
00:01:36.900 --> 00:01:43.640
And as π§ equals three π₯, ππ§ by ππ₯ is π by ππ₯ over three π₯, which is three.
00:01:44.450 --> 00:01:51.450
So altogether, ππ¦ by ππ₯ is six cos π§ times three which is 18 cos π§.
00:01:52.200 --> 00:01:56.170
And we donβt want ππ¦ by ππ₯ written in terms of some other variable π§.
00:01:56.410 --> 00:01:58.550
Weβd like it written in terms of π₯ if possible.
00:01:59.230 --> 00:02:06.470
Using the fact that π§ is three π₯, we see that ππ¦ by ππ₯ is 18 cos three π₯.
00:02:07.330 --> 00:02:09.260
And this is our final answer.
00:02:10.530 --> 00:02:20.080
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.