Find dx/dy 7cos(xy)=3x+8y

Problem

7*cos(x*y)=3*x+8*y

Solution

1. Differentiate both sides with respect to y treating x as a function of y

(d(7)*cos(x*y))/d(y)=d(3*x+8*y)/d(y)

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

−7*sin(x*y)⋅(d(x)*y)/d(y)=3d(x)/d(y)+8

3. Apply the product rule to the term (d(x)*y)/d(y)

−7*sin(x*y)⋅(x⋅1+yd(x)/d(y))=3d(x)/d(y)+8

4. Distribute the term −7*sin(x*y) into the parentheses.

−7*x*sin(x*y)−7*y*sin(x*y)d(x)/d(y)=3d(x)/d(y)+8

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

−7*y*sin(x*y)d(x)/d(y)−3d(x)/d(y)=8+7*x*sin(x*y)

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

d(x)/d(y)*(−7*y*sin(x*y)−3)=8+7*x*sin(x*y)

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

d(x)/d(y)=(8+7*x*sin(x*y))/(−7*y*sin(x*y)−3)

Final Answer

d(x)/d(y)=−(8+7*x*sin(x*y))/(7*y*sin(x*y)+3)

Want more problems? Check here!