Find dy/dx y^2+2y+3x=0

Problem

y2+2*y+3*x=0

Solution

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

d(y2)/d(x)+(d(2)*y)/d(x)+(d(3)*x)/d(x)=d(0)/d(x)

2. Apply the chain rule to the terms involving y and the power rule to the term involving x

2*yd(y)/d(x)+2d(y)/d(x)+3=0

3. Isolate the terms containing d(y)/d(x) on one side of the equation by subtracting 3 from both sides.

2*yd(y)/d(x)+2d(y)/d(x)=−3

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

d(y)/d(x)*(2*y+2)=−3

5. Solve for d(y)/d(x) by dividing both sides by the expression (2*y+2)

d(y)/d(x)=(−3)/(2*y+2)

Final Answer

d(y)/d(x)=−3/(2*y+2)

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