A lot of differentiation is about recognizing what kind of function you have in front of you. For instance, if you have a function with a power you may use the power rule or even the chain rule combined with the power rule. In that case, it only makes sense that you would use the product rule if you have the product of two functions. (Ok unless you can simplify first – then you need to make a decision as to whether or not it would be worth it!)
There is a pattern you can use to help yourself remember both the product and the quotient rule but it is a little different than the way a lot of textbooks tend to write it (don’t worry, it is the same). To take the derivative of a product, say 
Prime on the first term, prime on the second term
After you apply this formula, you can now look at the two derivatives in from of you (which are most likely more simple) and approach them as you would any other derivative. Also like any derivative, the difficulty can vary quite a bit. Take for instance a “basic” example:
Recognizing this as a product, I first apply the product rule by writing the product out twice and putting a prime on the first term of the first product and a prime on the second term of the second product.
Now I can take the two derivatives like usual (which you should notice are more simple than what we started with!) and get 
When this does get more complicated, it is mostly because the individual functions in the product are more complicated (think about it, in the end thats what you end up taking the derivative of). Composite functions will require the chain rule, or you might need the quotient rule, etc. In each case though, if you have a product that you can’t simplify (and maybe even if you can) you will apply the product rule and worry about what to do with it after that first step.
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