Answer
The difference quotient for the given function is $-6x-3h+1$
Work Step by Step
$f(x)=-3x^{2}+x-1$
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:
$f(x+h)=-3(x+h)^{2}+x+h-1=...$
$...=-3(x^{2}+2xh+h^{2})+x+h-1=...$
$...=-3x^{2}-6xh-3h^{2}+x+h-1$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{-3x^{2}-6xh-3h^{2}+x+h-1-(-3x^{2}+x-1)}{h}=...$
$...=\dfrac{-3x^{2}-6xh-3h^{2}+x+h-1+3x^{2}-x+1}{h}=...$
$...=\dfrac{-6xh-3h^{2}+h}{h}=...$
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 $-6x-3h+1$
