Question

use the formula f(x)=lim_{z→x} (f(z)-f(x))/(z - x) to find the derivative of the following function f(x)=4 + √(5x) f(x)=□ (type an exact answer, using radicals as needed.)

Explanation:

Step1: Find f(z)

$f(z)=4 + \sqrt{5z}$

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

$f^{\prime}(x)=\lim_{z
ightarrow x}\frac{(4 + \sqrt{5z})-(4+\sqrt{5x})}{z - x}=\lim_{z
ightarrow x}\frac{\sqrt{5z}-\sqrt{5x}}{z - x}$

Step3: Rationalize the numerator

Multiply the 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 the $(z - x)$ terms:
$f^{\prime}(x)=\lim_{z
ightarrow x}\frac{5}{\sqrt{5z}+\sqrt{5x}}=\frac{5}{2\sqrt{5x}}$

Answer:

$\frac{5}{2\sqrt{5x}}$