0

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?

Thor
  • 527
  • 1
    You will need to be more specific about the context. The "product rule" for example might be confused for something in elementary set theory and combinatorics stating that for finite sets $A$ and $B$ one has that $|A\times B|=|A|\times |B|$. Given your tags, this doesn't appear to be what you are referring to. – JMoravitz Jun 11 '18 at 02:34
  • 3
    Perhaps by "limit of product" you mean how $\lim\limits_{x\to c} f(x)\cdot g(x) = \lim\limits_{x\to c}f(x) \cdot \lim\limits_{x\to c} g(x)$ for suitable $f,g$ satisfying certain conditions. Perhaps by "product rule" you mean how $\frac{d}{dx}[f(x)g(x)] = \frac{d}{dx}[f(x)]\cdot g(x)+f(x)\cdot\frac{d}{dx}[g(x)]$. They are rather different. – JMoravitz Jun 11 '18 at 02:37

1 Answers1

3

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 $$h'(x)=f'(x)g(x)+f(x)g'(x).$$

Eric Wofsey
  • 342,377