How Do You Determine When to Use the Chain Rule or Product Rule in Calculus?

[duki]

I have a question more than a problem to answer.
I'm having a difficult time recognizing when to use the product rule and when to use the chain rule.

How do you recognize when to use each, especially when you have to use both in the same problem. Problems like y+x^4y^3-5x^6+3y^8-42=0 tend to mix me up.

 

[rocomath]

Product rule: x times y \frac{d}{dx}(xy)=xy'+y

Chain rule: 2x+2 raised to the power of 2 \frac{d}{dx}(2x+2)^2=2(2x+2)\cdot2