Answer
The difference quotient for $f\left( x \right)=3{{x}^{2}}-5x$ is \[6x+3h-5\]
Work Step by Step
The formula of the difference quotient for the function $f\left( x \right)$ is $\frac{f\left( x+h \right)-f\left( x \right)}{\left( x+h \right)-x}$
The given function is $f\left( x \right)=3{{x}^{2}}-5x$
Now, $f\left( x+h \right)=3{{\left( x+h \right)}^{2}}-5\left( x+h \right)$
$=3\left( {{x}^{2}}+2xh+{{h}^{2}} \right)-\left( 5x+5h \right)$
Therefore, $f\left( x+h \right)=3{{x}^{2}}+6xh+3{{h}^{2}}-5x-5h$
$f\left( x+h \right)-f\left( x \right)=3{{x}^{2}}+6xh+3{{h}^{2}}-5x-5h-\left( 3{{x}^{2}}-5x \right)$
$=3{{x}^{2}}+6xh+3{{h}^{2}}-5x-5h-3{{x}^{2}}+5x$
$=6xh+3{{h}^{2}}-5h$
Therefore, $\frac{f\left( x+h \right)-f\left( x \right)}{\left( x+h \right)-x}=\frac{6xh+3{{h}^{2}}-5h}{x+h-x}$
$=\frac{6xh+3{{h}^{2}}-5h}{h}$
$=\frac{h\left( 6x+3h-5 \right)}{h}$
Therefore, \[\frac{f\left( x+h \right)-f\left( x \right)}{\left( x+h \right)-x}=\,\,6x+3h-5\]
