find the difference quotient (f(x + h)-f(x))/...
find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=-4x² - x + 6 simplify your answer as much as possible. (f(x + h)-f(x))/h =
Answer
# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$:
\[
\begin{align*}
f(x + h)&=-4(x + h)^2-(x + h)+6\\
&=-4(x^{2}+2xh+h^{2})-x - h+6\\
&=-4x^{2}-8xh-4h^{2}-x - h+6
\end{align*}
\]
## Step2: Calculate $f(x + h)-f(x)$
\[
\begin{align*}
f(x + h)-f(x)&=(-4x^{2}-8xh-4h^{2}-x - h+6)-(-4x^{2}-x + 6)\\
&=-4x^{2}-8xh-4h^{2}-x - h+6 + 4x^{2}+x - 6\\
&=-8xh-4h^{2}-h
\end{align*}
\]
## Step3: Find the difference - quotient
\[
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{-8xh-4h^{2}-h}{h}\\
&=\frac{h(-8x-4h - 1)}{h}\\
&=-8x-4h - 1
\end{align*}
\]
# Answer:
$-8x-4h - 1$