What is the equation of the normal line of f(x)=1−ln⁡xx at x=1?

Given f(x)=1−ln⁡xx and x=1find f(1)f(1)=1−ln⁡(1)1f(1)=1Use point (1,1)Find the slope of the Normal Line at (1,1)slope =-1/f'(1)f′(x)=0+(−1)⋅x⋅1x−ln⁡x⋅1x2f′(1)=(−1)⋅1−ln⁡112f'(1)=-1slope m=−1−1=1Use point slope formy−y1=m(x−x1)y-1=1(x-1)y=x

Expert Answer to "What is the equation of the normal line..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Harrison Reeves

Answered question

2022-04-08

What is the equation of the normal line of

f (x) = 1 - \frac{\ln x}{x}

at x=1?

Answer & Explanation

tempur8x43

Beginner2022-04-09Added 16 answers

Given

f (x) = 1 - \frac{\ln x}{x}

and x=1
find f(1)

f (1) = 1 - \frac{\ln (1)}{1}

f(1)=1
Use point (1,1)
Find the slope of the Normal Line at (1,1)
slope =-1/f'(1)

f^{'} (x) = 0 + (- 1) \cdot \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^{2}}

f^{'} (1) = (- 1) \cdot \frac{1 - \ln 1}{1^{2}}

f'(1)=-1
slope

m = - \frac{1}{- 1} = 1

Use point slope form

y - y_{1} = m (x - x_{1})

y-1=1(x-1)
y=x

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