Question

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.)

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:

\(\frac{\sqrt{5}}{2\sqrt{x}}\)