the derivative as a function
question
completed: 0 of 6 | my score: 0/6 pts (0%
use the formula f(x)=\lim_{z\to x}\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.
f(x)=\frac{3}{x + 10}
f(x)=\square

Question

Accepted Answer

# Explanation:
## Step1: Find f(z)
Given $f(x)=\frac{3}{x + 10}$, then $f(z)=\frac{3}{z + 10}$.
## Step2: Substitute f(z) and f(x) into the derivative formula
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{f(z)-f(x)}{z - x}\
&=\lim_{z
ightarrow x}\frac{\frac{3}{z + 10}-\frac{3}{x + 10}}{z - x}\
&=\lim_{z
ightarrow x}\frac{\frac{3(x + 10)-3(z + 10)}{(z + 10)(x + 10)}}{z - x}\
&=\lim_{z
ightarrow x}\frac{3x+30 - 3z-30}{(z - x)(z + 10)(x + 10)}\
&=\lim_{z
ightarrow x}\frac{3x-3z}{(z - x)(z + 10)(x + 10)}\
&=\lim_{z
ightarrow x}\frac{-3(z - x)}{(z - x)(z + 10)(x + 10)}
\end{align*}
$$
## Step3: Simplify the limit
Cancel out the $(z - x)$ terms:
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{-3}{(z + 10)(x + 10)}\
&=\frac{-3}{(x + 10)(x + 10)}\
&=-\frac{3}{(x + 10)^2}
\end{align*}
$$
# Answer:
$-\frac{3}{(x + 10)^2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find f(z)

Step2: Substitute f(z) and f(x) into the derivative formula

Step3: Simplify the limit

Answer: