the derivative as a function
question
completed: 3 of 6 | my score: 3/6 pts (50%)
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)=2+\sqrt{5x}
f(x)= (type an exact answer, using radicals as needed.)

Question

Accepted Answer

# Explanation:
## Step1: Find f(z)
Given $f(x)=2 + \sqrt{5x}$, then $f(z)=2+\sqrt{5z}$.
## Step2: Substitute into derivative formula
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{f(z)-f(x)}{z - x}\
&=\lim_{z
ightarrow x}\frac{(2+\sqrt{5z})-(2+\sqrt{5x})}{z - x}\
&=\lim_{z
ightarrow x}\frac{\sqrt{5z}-\sqrt{5x}}{z - x}
\end{align*}
$$
## Step3: Rationalize the numerator
Multiply numerator and denominator by $\sqrt{5z}+\sqrt{5x}$
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{(\sqrt{5z}-\sqrt{5x})(\sqrt{5z}+\sqrt{5x})}{(z - x)(\sqrt{5z}+\sqrt{5x})}\
&=\lim_{z
ightarrow x}\frac{5z - 5x}{(z - x)(\sqrt{5z}+\sqrt{5x})}\
&=\lim_{z
ightarrow x}\frac{5(z - x)}{(z - x)(\sqrt{5z}+\sqrt{5x})}
\end{align*}
$$
## Step4: Simplify and find the limit
Cancel out $(z - x)$ terms:
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{5}{\sqrt{5z}+\sqrt{5x}}\
&=\frac{5}{\sqrt{5x}+\sqrt{5x}}\
&=\frac{5}{2\sqrt{5x}}=\frac{\sqrt{5}}{2\sqrt{x}}
\end{align*}
$$
# Answer:
$\frac{\sqrt{5}}{2\sqrt{x}}$

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Question

Explanation:

Step1: Find f(z)

Step2: Substitute into derivative formula

Step3: Rationalize the numerator

Step4: Simplify and find the limit

Answer: