If 𝑦 equals sin of four 𝑥 minus eight plus cos of eight 𝑥 plus six, find d𝑦 by d𝑥.
We’ve been asked to find the first derivative of this function, 𝑦, which is the sum of two trigonometric functions. We can, therefore, recall a helpful cycle which we can use for finding the derivatives of sine and cosine functions. 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. And then, we go around the cycle again. As integration is the reverse process of differentiation, we can go around this cycle in the other direction in order to find the integrals of sine and cosine functions. Remembering that we must add a constant of integration if our integral is indefinite.
It’s also important that we remember that we can only use these rules for differentiation if we’re working in radian measure. In the question we’ve been given, we can see that it isn’t just sin 𝑥 and cos 𝑥. It’s sin of a function of 𝑥, four 𝑥 minus eight, and cos of a different function of 𝑥, eight 𝑥 plus six, both of which are linear functions. We recall then a second set of general rules. The derivative with respect to 𝑥 of sin of 𝑎𝑥 plus 𝑏 is 𝑎 cos 𝑎𝑥 plus 𝑏. And the derivative with respect to 𝑥 of cos 𝑎𝑥 plus 𝑏 is negative 𝑎 sin 𝑎𝑥 plus 𝑏. We have a multiplicative factor of 𝑎 which is the derivative of the function inside the brackets.
These results are actually initiative of an even more general result. Which is that the derivative with respect to 𝑥 of sin of some function 𝑓 of 𝑥 is equal to 𝑓 prime of 𝑥 multiplied by cos of 𝑓 of 𝑥. We multiply by the derivative of the function whose sine we’re taking. And there is, of course, an equivalent rule for finding the derivative of cos of some function 𝑓 of 𝑥. We can prove these results using the chain rule. But we don’t need to do that here. We’ll just quote the standard results for the derivatives of sin 𝑎𝑥 plus 𝑏 and cos 𝑎𝑥 plus 𝑏.
We also recall that in order to find the derivative of a sum, we can just take the sum of the derivatives. We just differentiate each term separately. So differentiating sin of four 𝑥 minus eight, first of all. This will give four multiplied by cos of four 𝑥 minus eight. Then, we add the derivative of cos eight 𝑥 plus six. Well, that’s going to be negative eight multiplied by sin of eight 𝑥 plus six. We can write these terms in either order. And it’s perhaps more usual to see sine terms written before cosine terms.
So by recalling the standard results for the derivatives of sine and cosine functions, we’ve found that if 𝑦 is equal to sin of four 𝑥 minus eight plus cos of eight 𝑥 plus six. Then its first derivative with respect to 𝑥, d𝑦 by d𝑥, is equal to negative eight sin eight 𝑥 plus six plus four cos four 𝑥 minus eight.