Video: Differentiating a Combination of Trigonometric Functions

If 𝑦 = 6 cos 4π‘₯ + 2 sin 2π‘₯, find d𝑦/dπ‘₯.

02:32

Video Transcript

If 𝑦 equals six cos four π‘₯ plus two sin two π‘₯, find d𝑦 by dπ‘₯.

In this question, we’re asked to find the derivative of a sum of two trigonometric functions. So, we need to recall the rules for differentiating these. 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‘₯. Although, we must remember that these rules are only true if the angle is measured in radians.

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. So, we also need to recall how to differentiate sine and cosine functions of this type. We can quote further standard results. For a constant π‘Ž, the derivative with respect to π‘₯ of sin π‘Žπ‘₯ is π‘Ž cos π‘₯. And the derivative with respect to π‘₯ cos π‘Žπ‘₯ is negative π‘Ž sin π‘Žπ‘₯. We just have an extra factor of π‘Ž in our derivatives. These results could be proved using the chain rule if we wish.

Now, notice that we also have multiplicative constants in front of each term. We have a six in the first term and a two in the second. But we know that multiplying by a constant just means that the derivative will also be multiplied by the same constant. So, we can say that the derivative of 𝑦 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‘₯. And we can use our standard results to find each of these derivatives.

Applying the second rule for the derivative with respect to π‘₯ of cos π‘Žπ‘₯, we have that the derivative with respect to π‘₯ of cos four π‘₯ is negative four sin four π‘₯. And then, applying our first rule for the derivative of sin π‘Žπ‘₯, we have that the derivative with respect to π‘₯ of sin two π‘₯ is two cos two π‘₯. So, d𝑦 by dπ‘₯ is equal to six multiplied by negative four sin four π‘₯ plus two multiplied by two cos two π‘₯. Simplifying the constants, and we have our answer. d𝑦 by dπ‘₯ is equal to negative 24 sin four π‘₯ plus four cos two π‘₯. And again, remember that this result is only true when π‘₯ is measured in radians.

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