Answer
The difference quotient for the given function is $6x+3h+1$
Work Step by Step
$f(x)=3x^{2}+x+5$
Find the difference quotient $\dfrac{f(x+h)-f(x)}{h}$
Start by finding $f(x+h)$. Substitute $x$ by $x+h$ in $f(x)$ and simplify:
$f(x+h)=3(x+h)^{2}+x+h+5=...$
$...=3(x^{2}+2xh+h^{2})+x+h+5=...$
$...=3x^{2}+6xh+3h^{2}+x+h+5$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{3x^{2}+6xh+3h^{2}+x+h+5-(3x^{2}+x+5)}{h}=...$
$...=\dfrac{3x^{2}+6xh+3h^{2}+x+h+5-3x^{2}-x-5}{h}=...$
$...=\dfrac{6xh+3h^{2}+h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(6x+3h+1)}{h}=6x+3h+1$
The difference quotient for the given function is $6x+3h+1$
