If 𝑦 equals six cos four 𝑥 plus two sin two 𝑥, find d𝑦 by d𝑥.
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 𝑥 is cos 𝑥, the derivative of cos 𝑥 is negative sin 𝑥, the derivative of negative sin 𝑥 is negative cos 𝑥, and the derivative of negative cos 𝑥 is sin 𝑥. 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 𝑥; we have four 𝑥 in the first term, and we have two 𝑥 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 𝑎, the derivative with respect to 𝑥 of sin 𝑎𝑥 is 𝑎 cos 𝑥. And the derivative with respect to 𝑥 cos 𝑎𝑥 is negative 𝑎 sin 𝑎𝑥. We just have an extra factor of 𝑎 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 𝑥 will be equal to six multiplied by the derivative of cos four 𝑥 with respect to 𝑥 plus two multiplied by the derivative of sin two 𝑥 with respect to 𝑥. And we can use our standard results to find each of these derivatives.
Applying the second rule for the derivative with respect to 𝑥 of cos 𝑎𝑥, we have that the derivative with respect to 𝑥 of cos four 𝑥 is negative four sin four 𝑥. And then, applying our first rule for the derivative of sin 𝑎𝑥, we have that the derivative with respect to 𝑥 of sin two 𝑥 is two cos two 𝑥. So, d𝑦 by d𝑥 is equal to six multiplied by negative four sin four 𝑥 plus two multiplied by two cos two 𝑥. Simplifying the constants, and we have our answer. d𝑦 by d𝑥 is equal to negative 24 sin four 𝑥 plus four cos two 𝑥. And again, remember that this result is only true when 𝑥 is measured in radians.