# Find the difference quotient of f, that is, find (f(x+h)−f(x))/h , h≠0, for each function. Be sure to simplify. f(x) = 1/(x+3)

Analyzing functions
Find the difference quotient of f, that is, find $$\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}−{f{{\left({x}\right)}}}}}{{h}},{h}≠{0}$$, for each function. Be sure to simplify. $$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}$$
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}$$
$$\displaystyle{f{{\left({x}+{h}\right)}}}–{f{{\left({x}\right)}}}\frac{{)}}{{h}}=\frac{{\frac{{1}}{{{x}+{h}+{3}}}–\frac{{1}}{{{x}+{3}}}}}{{h}}=\frac{{\frac{{{x}+{3}–{\left({x}+{3}+{h}\right)}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}}}{{h}}={\left(\frac{{{x}+{3}–{x}–{3}–{h}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}\right)}{\left(\frac{{1}}{{h}}\right)}={\left(\frac{{-{h}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}\right)}{\left(\frac{{1}}{{h}}\right)}=\frac{{-{1}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}$$