WEBVTT
00:00:01.860 --> 00:00:07.190
If π¦ equals six cos four π₯ plus two sin two π₯, find dπ¦ by dπ₯.
00:00:08.810 --> 00:00:13.720
In this question, weβre asked to find the derivative of a sum of two trigonometric functions.
00:00:13.900 --> 00:00:17.130
So, we need to recall the rules for differentiating these.
00:00:18.590 --> 00:00:31.180
Our most basic rules tell us that the derivative of sin π₯ is cos π₯, the derivative of cos π₯ is negative sin π₯, the derivative of negative sin π₯ is negative cos π₯, and the derivative of negative cos π₯ is sin π₯.
00:00:31.350 --> 00:00:36.110
Although, we must remember that these rules are only true if the angle is measured in radians.
00:00:37.500 --> 00:00:45.950
We can see, though, that the arguments in our function are not just π₯; we have four π₯ in the first term, and we have two π₯ in the second.
00:00:46.130 --> 00:00:50.990
So, we also need to recall how to differentiate sine and cosine functions of this type.
00:00:52.510 --> 00:00:54.620
We can quote further standard results.
00:00:54.710 --> 00:01:00.180
For a constant π, the derivative with respect to π₯ of sin ππ₯ is π cos π₯.
00:01:00.370 --> 00:01:05.070
And the derivative with respect to π₯ cos ππ₯ is negative π sin ππ₯.
00:01:05.340 --> 00:01:08.600
We just have an extra factor of π in our derivatives.
00:01:09.860 --> 00:01:12.840
These results could be proved using the chain rule if we wish.
00:01:13.150 --> 00:01:17.150
Now, notice that we also have multiplicative constants in front of each term.
00:01:17.150 --> 00:01:20.100
We have a six in the first term and a two in the second.
00:01:20.480 --> 00:01:27.190
But we know that multiplying by a constant just means that the derivative will also be multiplied by the same constant.
00:01:28.540 --> 00:01:41.260
So, we can say that the derivative of π¦ with respect to π₯ will be equal to six multiplied by the derivative of cos four π₯ with respect to π₯ plus two multiplied by the derivative of sin two π₯ with respect to π₯.
00:01:41.520 --> 00:01:45.070
And we can use our standard results to find each of these derivatives.
00:01:46.860 --> 00:01:57.400
Applying the second rule for the derivative with respect to π₯ of cos ππ₯, we have that the derivative with respect to π₯ of cos four π₯ is negative four sin four π₯.
00:01:57.850 --> 00:02:07.600
And then, applying our first rule for the derivative of sin ππ₯, we have that the derivative with respect to π₯ of sin two π₯ is two cos two π₯.
00:02:09.010 --> 00:02:16.060
So, dπ¦ by dπ₯ is equal to six multiplied by negative four sin four π₯ plus two multiplied by two cos two π₯.
00:02:17.220 --> 00:02:25.170
Simplifying the constants, and we have our answer. dπ¦ by dπ₯ is equal to negative 24 sin four π₯ plus four cos two π₯.
00:02:25.400 --> 00:02:30.420
And again, remember that this result is only true when π₯ is measured in radians.