Find the slope of the line tangent to the

graph of ln (xy) - 2x=0 when x = - 1

answer choices ...

1. slope = 3/2e^-2

2. slope = 3/2e^2

3. slope = 3e^-2

4. slope = - 3e^2

5. slope = - 3/2e^2

6. slope = - 3e^-2

Ln (xy) - 2x = 0

slope of the tangent line = derivative of the function

[ln (xy) ]' = [2x]'

[1 / (xy) ] [y + xy'] = 2

y + xy' = 2 (xy)

xy' = 2xy - y = y (2x-1)

y' = y (2x-1) / x

Now use x = - 1 to find y and after to find y'

ln (xy) = 2x

x=-1

ln (-y) = - 2

-y = e^-2

y = - e^-2

y' = [-e^-2][2 (-1) - 1] / (-1) = [e^-2] (-2-1) = [e^-2] (-3) = - 3e^-2

Answer: option 6. from the list