derivative, the difference between "the limit of product" and the "product rule"

Question

I'm currently studying derivative and I'm really confused by the concept of "the limit of product" and the "product rule". Are they the two different name for the same concept or are they completely different?

Answer 1

Well, I don't know how these terms are used in your course, but they usually have different meanings. The phrase "limit of product" would refer to limits of products in general. If $f$ and $g$ are any two functions, you can consider a limit $$\lim_{x\to a}f(x)g(x).$$ In particular, there is a theorem that if $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ both exist, then $\lim_{x\to a}f(x)g(x)$ exists and $$\lim_{x\to a}f(x)g(x)=\left(\lim_{x\to a}f(x)\right)\cdot \left(\lim_{x\to a}f(x)\right).$$ Or, briefly, "the limit of a product is the product of the limits".

The "product rule", on the other hand, usually refers to something else, a rule about derivatives (not limits) of functions. Specifically, it says that if $f$ and $g$ are differentiable functions and $h$ is the function $h(x)=f(x)g(x)$, then $$h'(x)=f'(x)g(x)+f(x)g'(x).$$