Suppose that f\displaystyle {f} is a function given as f(x)=3x4\displaystyle {f{{\left({x}\right)}}}=-{3}{x}-{4}.

Simplify the expression f(x+h)\displaystyle {f{{\left({x}+{h}\right)}}}.
f(x+h)\displaystyle {f{{\left({x}+{h}\right)}}} =  

Simplify the difference quotient, f(x+h)f(x)h\displaystyle \frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}.
f(x+h)f(x)h\displaystyle \frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}} =  

The derivative of the function at x\displaystyle {x} is the limit of the difference quotient as h approaches zero.
f(x)=limh0\displaystyle {f}'{\left({x}\right)}=\lim_{{{h}\rightarrow{0}}} f(x+h)f(x)h\displaystyle \frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}} =