Find dy/dx e^ycos(x)=2+sin(xy)

Problem

ey*cos(x)=2+sin(x*y)

Solution

1. Differentiate both sides with respect to x using the chain rule and product rule, treating y as a function of x

(d(ey)*cos(x))/d(x)=d(2+sin(x*y))/d(x)

2. Apply the product rule to the left side and the chain rule to the right side.

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

3. Compute the derivatives of the individual terms, applying the product rule to x*y and the chain rule to ey

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

4. Expand the terms on the right side to isolate the terms containing d(y)/d(x)

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

5. Group all terms involving d(y)/d(x) on one side and the remaining terms on the other side.

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

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

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

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

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

Final Answer

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

Want more problems? Check here!