question let (f(x)=-3x^{2}-6x - 13). find (f(3)) using the definition of a derivative. provide your answer below: (f(3)=square)

Question

Accepted Answer

# Explanation:
## Step1: Recall derivative definition
The definition of the derivative of a function $y = f(x)$ at a point $x=a$ is $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here $a = 3$ and $f(x)=-3x^{2}-6x - 13$. First, find $f(3 + h)$ and $f(3)$.
$$
\begin{align*}
f(3 + h)&=-3(3 + h)^{2}-6(3 + h)-13\
&=-3(9 + 6h+h^{2})-18-6h - 13\
&=-27-18h-3h^{2}-18 - 6h-13\
&=-3h^{2}-24h - 58
\end{align*}
$$
$$
f(3)=-3\times3^{2}-6\times3-13=-3\times9 - 18-13=-27-18 - 13=-58
$$
## Step2: Substitute into derivative - formula
$$
\begin{align*}
f^{\prime}(3)&=\lim_{h
ightarrow0}\frac{f(3 + h)-f(3)}{h}\
&=\lim_{h
ightarrow0}\frac{-3h^{2}-24h - 58-(-58)}{h}\
&=\lim_{h
ightarrow0}\frac{-3h^{2}-24h - 58 + 58}{h}\
&=\lim_{h
ightarrow0}\frac{-3h^{2}-24h}{h}
\end{align*}
$$
## Step3: Simplify the expression
$$
\begin{align*}
\lim_{h
ightarrow0}\frac{-3h^{2}-24h}{h}&=\lim_{h
ightarrow0}\frac{h(-3h - 24)}{h}\
&=\lim_{h
ightarrow0}(-3h - 24)
\end{align*}
$$
## Step4: Evaluate the limit
As $h
ightarrow0$, we substitute $h = 0$ into $-3h - 24$. So $f^{\prime}(3)=-24$.
# Answer:
$-24$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall derivative definition

Step2: Substitute into derivative - formula

Step3: Simplify the expression

Step4: Evaluate the limit

Answer: