Find dy/dx if ln(xy) = x+y

Question

Which of the following statements about dy/dx if ln(xy) = x+y is true?

a)dy/dx = 1/x -1 / 1 - 1/y

b)dy/dx = 1/x -1/y + 1

c)dy/dx = 1/x -1 / 1/y - 1

d)dy/dx = 1/x + 1/y -1

Answer 1

ln(xy) = x+y

Take derivative of both sides

(1/(xy))(y+xdy/dx) = 1 + dy/dx

y+xdy/dx = xy + xydy/dx

xdy/dx-xydy/dx = xy-y

dy/dx = (xy-y)/(x-xy) = (y/x)(x-1)/(1-y) = (1-1/x)/(1/y-1) = (1/x-1)/(1-1/y)

So the answer is a)