Find dy/dx xe^y+1=xy

Problem

x*ey+1=x*y

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x and applying the product rule to both x*ey and x*y

(d(x)*ey)/d(x)+d(1)/d(x)=(d(x)*y)/d(x)

2. Apply the product rule and the chain rule to the left side, where (d(x)*ey)/d(x)=ey+x*eyd(y)/d(x) and the derivative of the constant 1 is 0

ey+x*eyd(y)/d(x)+0=(d(x)*y)/d(x)

3. Apply the product rule to the right side, where (d(x)*y)/d(x)=y+xd(y)/d(x)

ey+x*eyd(y)/d(x)=y+xd(y)/d(x)

4. Group the terms containing d(y)/d(x) on one side of the equation and the remaining terms on the other side.

x*eyd(y)/d(x)−xd(y)/d(x)=y−ey

5. Factor out the common term d(y)/d(x) from the left side.

d(y)/d(x)*(x*ey−x)=y−ey

6. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(y−ey)/(x*ey−x)

Final Answer

d(y)/d(x)=(y−ey)/(x*ey−x)

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