Find dy/dx x^3-xy+y^2=4

Problem

x3−x*y+y2=4

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(x3)/d(x)−(d(x)*y)/d(x)+d(y2)/d(x)=d(4)/d(x)

2. Apply the product rule to the term x*y and the power rule with the chain rule to the other terms.

3*x2−(xd(y)/d(x)+y⋅1)+2*yd(y)/d(x)=0

3. Distribute the negative sign and group all terms containing d(y)/d(x) on one side of the equation.

3*x2−xd(y)/d(x)−y+2*yd(y)/d(x)=0

2*yd(y)/d(x)−xd(y)/d(x)=y−3*x2

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

d(y)/d(x)*(2*y−x)=y−3*x2

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

d(y)/d(x)=(y−3*x2)/(2*y−x)

Final Answer

d(y)/d(x)=(y−3*x2)/(2*y−x)

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