By use implicit derivative find of Inxy = e*y

Given:

ln(xy) = exp(x + y)

(where exp(x) denote the exponential function, eˣ )

Differentiate both sides with respect to x :

d/dx [ln(xy)] = d/dx [exp(x + y)]

Chain rule:

1/(xy) • d/dx [xy] = exp(x + y) • d/dx [x + y]

Product rule on the left, and sum rule on the right. Keep in mind that y = y(x) :

1/(xy) (y + x dy/dx) = exp(x + y) (1 + dy/dx)

Solve for dy/dx :

(y + x • dy/dx) / (xy) = exp(x + y) (1 + dy/dx)

1/x + 1/y • dy/dx = exp(x + y) + exp(x + y) • dy/dx

1/y • dy/dx - exp(x + y) • dy/dx = exp(x + y) - 1/x

(1/y - exp(x + y)) • dy/dx = exp(x + y) - 1/x

dy/dx = (exp(x + y) - 1/x) / (1/y - exp(x + y))

Multiply the right side by x/x and y/y to eliminate the rational terms:

dy/dx = (xy exp(x + y) - y) / (x - xy exp(x + y))