Question

difference quotient

Answer 1

if this is about limits:

\(\large\color{black}{\displaystyle\lim_{x \rightarrow ~a}\left[~f(x)-g(x) ~\right]=\left[\displaystyle\lim_{x \rightarrow ~a}f(x)~-~\displaystyle\lim_{x \rightarrow ~a}g(x) ~\right]}\)


\(\large\color{black}{\displaystyle\lim_{x \rightarrow ~a}\left[~\frac{f(x)}{g(x)} ~\right]=\left[~\frac{\displaystyle\lim_{x \rightarrow ~a}f(x)}{\displaystyle\lim_{x \rightarrow ~a}g(x)} ~\right]}\)

Answer 2

if this is about derivatives:

\(\Large\color{black}{\frac{d}{dx}\left[~f(x)-g(x) ~\right]=\frac{d}{dx}f(x)-\frac{d}{dx}g(x)}\)


\(\Large\color{black}{\frac{d}{dx}\left[~\frac{f(x)}{g(x)} ~\right]=\frac{g(x)~f'(x)-f(x)~g'(x)}{[~g(x)~]^2} }\)

Answer 3

If this is about integrals, then there is no quotient rule, but there is a difference:

\(\large\color{black}{\displaystyle\int\limits_{~}^{~}\left[~f(x)-g(x)~ \right]~dx=\displaystyle\int\limits_{~}^{~}f(x)~dx~-~\displaystyle\int\limits_{~}^{~}g(x)~dx}\)

Answer 4

@SolomonZelman the difference quotient has more to do with average rate of change than instantaneous rate of change (derivative).

Answer 5

no I was thinking that the poster asked for some rules, being that this
question is in calculus, this is what I posted. Be this question from just
mathematics I would never do this.

You would also agree though, that this question
doesn't have too much info :)