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

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Answer 1 · 2018-02-18T10:33:09

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

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Answer 2 · 2018-02-18T10:48:32

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#

#"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#