What is the equation of the tangent line of f(x)=ln(x+3)/x+7x at x=3?

2 Answers
Feb 18, 2018

y=(1-2ln6)/18x+(4ln6+125)/6

Explanation:

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Feb 18, 2018

Equation of the tangent line passing through the point P-=(3,21.085) with slope m=-0.147 is given by
0.147x+y-21.526=0

Explanation:

"We need to find the equation of the tangent line in the form "

px+qy+c=0
Given:

f(x)=ln(x+3)/(7x)+7x
Let y=f(x)
y=ln(x+3)/(7x)+7x

At x=3, f(x)=f(3)=ln(3+3)/(7xx3)+7xx3=21.085

Multiplying the functin with x

xy=1/7ln(x+3)+7x^2

xy=7x^2+1/7ln(x+3)

Differentiating wrt x on both sides

xy'+y=1/7(1/(x+3))+14x

Substituting for x and y

3y'+21.085=1/7(1/(3+3))+14xx3

Dividing throughout by 3
y'+21.085=1/42+42, y'=1/42+42-21.085=20.938

y'+21.085=20.938

y'=-0.147
Slope of the tangent line is m=-0.147

Equation of the tangent line passing through the point P-=(3,21.085) with slope m=-0.147 is given by

(y-21.085)/(x-3)=-0.147

y-21.085=-0.147(x-3)
y-21.085=-0.147x+0.440
0.147x+y-21.085-0.440=0

Equation of the tangent line passing through the point P-=(3,21.085) with slope m=-0.147 is given by
0.147x+y-21.526=0