use the function below to answer parts (a)-(c).\n\n$f(x)=\\frac{4}{x}$\n\n(a) use the formal definition to…

use the function below to answer parts (a)-(c).\n\n$f(x)=\\frac{4}{x}$\n\n(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 5$.\n(b) find $f(5)$ and find the equation of the normal line at the point $(5, f(5))$.\n(c) graph $y = f(x)$ and the tangent line at the point $(5, f(5))$ in the same coordinate system.\n\n(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 $x = 5$.\n\n$f(5)=\\lim_{h\\to0}\\frac{\\square-\\left\\frac{4}{x}\\right}{h}$

use the function below to answer parts (a)-(c).\n\n$f(x)=\\frac{4}{x}$\n\n(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 5$.\n(b) find $f(5)$ and find the equation of the normal line at the point $(5, f(5))$.\n(c) graph $y = f(x)$ and the tangent line at the point $(5, f(5))$ in the same coordinate system.\n\n(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 $x = 5$.\n\n$f(5)=\\lim_{h\\to0}\\frac{\\square-\\left\\frac{4}{x}\\right}{h}$

Answer

Explanation:

Step1: Substitute (x = 5) into (f(x)) and (f(x + h))

(f(5)=\frac{4}{5}), (f(5 + h)=\frac{4}{5 + h}) So (f^{\prime}(5)=\lim_{h\rightarrow0}\frac{\frac{4}{5 + h}-\frac{4}{5}}{h})

Step2: Simplify the numerator

(\frac{4}{5 + h}-\frac{4}{5}=\frac{4\times5-4\times(5 + h)}{5(5 + h)}=\frac{20-20 - 4h}{5(5 + h)}=\frac{-4h}{5(5 + h)})

Step3: Substitute the simplified numerator back into the limit

(f^{\prime}(5)=\lim_{h\rightarrow0}\frac{\frac{-4h}{5(5 + h)}}{h}=\lim_{h\rightarrow0}\frac{-4h}{5h(5 + h)})

Step4: Cancel out (h)

(f^{\prime}(5)=\lim_{h\rightarrow0}\frac{-4}{5(5 + h)})

Step5: Evaluate the limit

As (h\rightarrow0), (f^{\prime}(5)=\frac{-4}{5\times5}=-\frac{4}{25})

Answer:

(f^{\prime}(5)=-\frac{4}{25})