Usually any good calc textbook will walk you through the process. It's rather mechanical. If you have a function inside of another function, you differentiate the outside one and multiply it by the inside one. With the power rule, you are still using the chain rule without knowing it. For example, in differentiating the function
f(x) = x^2, you take the derivative of the "outside" (x^2) times the derivative of the "inside" (x) yielding f(x) = 2x*(1) . You multiply by one since the chain rule told you to multiply by the derivative of the inside function. I'm not sure if I answered your question, but you'd be better off taking a peek at a good calc textbook (try Stewarts), and working a few problems until you get it. It's really a skill you need to practice to understand how the rules apply.
If the equation is in the form f(x)=ax^n, then you can use the power rule. If it is a composite function of some form, you can use the chain rule to keep it simple. f(x)=a(x+3)^n can be expanded out and differentiated with the power rule, but it's much easier to use chain.
Although I believe that the power rule is more a derivation of first principles as opposed to application of the chain rule, you can use chain if you want to...but it's so much easier to use anx^n-1 (Power rule).