Find dy/dx e^ysin(x)=x+xy

Problem

ey*sin(x)=x+x*y

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

2. Apply the product rule to the left side, where the derivative of ey is eyd(y)/d(x) by the chain rule.

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

3. Expand the derivatives on both sides, applying the product rule to ey*sin(x) and x*y

ey*cos(x)+sin(x)*eyd(y)/d(x)=1+y+xd(y)/d(x)

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

sin(x)*eyd(y)/d(x)−xd(y)/d(x)=1+y−ey*cos(x)

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

d(y)/d(x)*(ey*sin(x)−x)=1+y−ey*cos(x)

6. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(1+y−ey*cos(x))/(ey*sin(x)−x)

Final Answer

d(y)/d(x)=(1+y−ey*cos(x))/(ey*sin(x)−x)

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