Find dy/dx e^ycos(x)=6+sin(xy)

Problem

ey*cos(x)=6+sin(x*y)

Solution

1. Differentiate both sides with respect to x using the product rule and the chain rule.

(d(ey)*cos(x))/d(x)=d(6+sin(x*y))/d(x)

2. Apply the product rule to the left side and the chain rule to the right side, remembering that y is a function of x

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

3. Distribute the cos(x*y) term on the right side of the equation.

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

4. Group all terms containing 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)

5. 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)

6. Solve for d(y)/d(x) 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!