Find dy/dx xe^y=y-1

Problem

x*ey=y−1

Solution

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

(d(x)*ey)/d(x)=d(y−1)/d(x)

2. Apply the product rule to the term x*ey which gives ey⋅d(x)/d(x)+x⋅d(ey)/d(x)

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

3. Group the terms containing d(y)/d(x) on one side of the equation to prepare for factoring.

ey=d(y)/d(x)−x*eyd(y)/d(x)

4. Factor out the common term d(y)/d(x) from the right side.

ey=d(y)/d(x)*(1−x*ey)

5. Solve for the derivative by dividing both sides by the expression (1−x*ey)

d(y)/d(x)=(ey)/(1−x*ey)

Final Answer

d(y)/d(x)=(ey)/(1−x*ey)

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