How do I differentiate a trigonometric function for something that is not just a single variable (e.g. d/dx (sin(3x))?

In order to differentiate a trig function with a term in front of the variable you are differentiating, you must use the chain rule. For example d/dx (sin (3x)) becomes 3cos(3x) because you have to multiply the two differentials: 3 and cos (3x).