Question

use the function below to answer parts (a)-(c).
$f(x)=\frac{2}{x}$
(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 5$.
(b) find $f(5)$ and find the equation of the normal line at the point $(5, f(5))$.
(c) graph $y = f(x)$ and the tangent line at the point $(5, f(5))$ in the same coordinate system.
(a) the derivative of a function $f$ at $x$, denoted by $f(x)$, is $f(x)=\lim_{h\to0}\frac{f(x + h)-f(x)}{h}$ provided that the limit exists. use the definition of the derivative of $f$ at $x$ to find the derivative of the given function $f(x)$ when $x = 5$.
$f(5)=\lim_{h\to0}\frac{\square-\left\frac{2}{x}\
ight}{h}$

Explanation:

Step1: Substitute \(f(x)\) and \(f(x + h)\) into the derivative formula

Given \(f(x)=\frac{2}{x}\), then \(f(5)=\frac{2}{5}\) and \(f(5 + h)=\frac{2}{5 + h}\).

$$ f^{\prime}(5)=\lim_{h ightarrow0}\frac{\frac{2}{5 + h}-\frac{2}{5}}{h} $$

Step2: Simplify the numerator

$$
LATEXBLOCK0
$$

So \(f^{\prime}(5)=\lim_{h
ightarrow0}\frac{\frac{-2h}{5(5 + h)}}{h}\)

Step3: Simplify the fraction

$$
LATEXBLOCK1
$$

Step4: Evaluate the limit

$$
LATEXBLOCK2
$$

Answer:

\(f^{\prime}(5)=-\frac{2}{25}\)