the derivative as a function\nquestion\ncompleted: 3 of 6 | my score: 3/6 pts (50%)\nuse the formula…

the derivative as a function\nquestion\ncompleted: 3 of 6 | my score: 3/6 pts (50%)\nuse the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.\nf(x)=2+\\sqrt{5x}\nf(x)= (type an exact answer, using radicals as needed.)

the derivative as a function\nquestion\ncompleted: 3 of 6 | my score: 3/6 pts (50%)\nuse the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.\nf(x)=2+\\sqrt{5x}\nf(x)= (type an exact answer, using radicals as needed.)

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\rightarrow x}\frac{f(z)-f(x)}{z - x}\ &=\lim_{z\rightarrow x}\frac{(2+\sqrt{5z})-(2+\sqrt{5x})}{z - x}\ &=\lim_{z\rightarrow 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\rightarrow x}\frac{(\sqrt{5z}-\sqrt{5x})(\sqrt{5z}+\sqrt{5x})}{(z - x)(\sqrt{5z}+\sqrt{5x})}\ &=\lim_{z\rightarrow x}\frac{5z - 5x}{(z - x)(\sqrt{5z}+\sqrt{5x})}\ &=\lim_{z\rightarrow 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\rightarrow 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}})