How would you use implicit differentiation to find dy/dx of the problem: 7x^2 + 6ln(xy) = 14 ?? I have a test tomorrow morning and im lost.

Question

This is an implicit differentiation problem involving the dy/dx variable and I just can't figure out how to find the derivative of the 6lnxy part of the problem.

Answer 1

7x² + 6ln(xy) = 14

 

 

Differentiate both sides of equation using the chain rule. The single quotations indicate derivative.

 

[7x² + 6ln(xy)]' = [14]'

 

If y=ln(x), then the derivative of y = 1/x.  This means to find the derivative of the log argument, and divide that result by the argument.  So we will have

 

14x + 6((y + xy') / xy) = 0

 

 

Next, is to manipulate the equation so that we solve for y'.

 

Subtract 14x on both sides of equation.

 

6((y + xy') / xy) = -14x

 

 

Divide both sides of equation by 6.

 

(y + xy') / xy = (-7/3)x

 

 

Multiply both sides of equation by xy.

 

y + xy' = (-7/3)x²y

 

 

Subtract both sides of equation by y.

 

xy' = (-7/3)x²y - y

 

 

Divide both sides of equation by x.

 

y' = (-7/3)xy - (y/x)