Find dy/dx x^2y+y^2x=-2

Problem

x2*y+y2*x=−2

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the product rule to both terms on the left side.

d(x2*y+y2*x)/d(x)=d(−2)/d(x)

2. Apply the product rule to the first term x2*y and the second term y2*x

(x2d(y)/d(x)+yd(x2)/d(x))+(y2d(x)/d(x)+xd(y2)/d(x))=0

3. Compute the derivatives of the individual components, using the chain rule for terms involving y

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

4. Group the terms containing d(y)/d(x) on one side of the equation and move all other terms to the opposite side.

x2d(y)/d(x)+2*x*yd(y)/d(x)=−2*x*y−y2

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

d(y)/d(x)*(x2+2*x*y)=−(2*x*y+y2)

6. Solve for the derivative by dividing both sides by the expression (x2+2*x*y)

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)

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