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

Question

Emilia Aguilar · Accepted Answer

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

William Abdalla · Answer

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#

Emily Ammerman · Answer

undefined To find the equation of the tangent line of f(x) = ln(x+3)/(x+7x) at x=3, we need to find the derivative of f(x) and evaluate it at x=3.

First, let's find the derivative of f(x) using the quotient rule:

f'(x) = [(x+7x)(1/(x+3)) - ln(x+3)(1+7)] / (x+7x)^2

Simplifying this expression, we get:

f'(x) = [(8x+1)/(x+3)^2] - [ln(x+3)(8)] / (x+7x)^2

Now, let's evaluate f'(x) at x=3:

f'(3) = [(8(3)+1)/(3+3)^2] - [ln(3+3)(8)] / (3+7(3))^2

Simplifying this expression, we get:

f'(3) = [25/36] - [ln(6)(8)] / 169

Therefore, the slope of the tangent line at x=3 is f'(3) = [25/36] - [ln(6)(8)] / 169.

To find the equation of the tangent line, we use the point-slope form:

y - f(3) = f'(3)(x - 3)

Now, substitute the values of f(3) and f'(3) into the equation:

y - [ln(3+3)/(3+7(3))] = [25/36] - [ln(6)(8)] / 169 * (x - 3)

Simplifying this equation, we get the equation of the tangent line of f(x) at x=3.