Find dy/dx x^2y+xy^2=6

Problem

x2*y+x*y2=6

Solution

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

d(x2*y+x*y2)/d(x)=d(6)/d(x)

2. Apply the product rule to the first term x2*y which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x2)*y)/d(x)=x2d(y)/d(x)+y*(2*x)

3. Apply the product rule to the second term x*y2 remembering to use the chain rule for y2

(d(x)*y2)/d(x)=x*(2*yd(y)/d(x))+y(1)2

4. Combine the results and set the derivative of the constant 6 to zero.

x2d(y)/d(x)+2*x*y+2*x*yd(y)/d(x)+y2=0

5. Group the terms containing d(y)/d(x) on one side and move the remaining terms to the other side.

x2d(y)/d(x)+2*x*yd(y)/d(x)=−2*x*y−y2

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

d(y)/d(x)*(x2+2*x*y)=−2*x*y−y2

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

d(y)/d(x)=(−2*x*y−y2)/(x2+2*x*y)

Final Answer

d(y)/d(x)=−(2*x*y+y2)/(x2+2*x*y)

Want more problems? Check here!