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This question is for the product rule proof:

a. [f(x+h)g(x+h)-f(x)g(x)]/h

->

b. [f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)]/h

->

c. [(f(x+h)-f(x))g(x+h) + f(x)(g(x+h)-g(x))]/h

I understand how in b. -/+f(x)g(x+h) are pulled out of thin air, but I'm missing how b. is modified into c. Is there a simple explanation for that?

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Yes: factor g(x+h) out of the first two terms of the numerator, and factor f(x) out of the last two terms of the numerator.