Find dy/dx x^3-3x^2y+2xy^2=12

Problem

x3−3*x2*y+2*x*y2=12

Solution

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

d(x3)/d(x)−(d(3)*x2*y)/d(x)+(d(2)*x*y2)/d(x)=d(12)/d(x)

2. Apply the product rule to the terms 3*x2*y and 2*x*y2

3*x2−(3*x2d(y)/d(x)+6*x*y)+(2*x⋅2*yd(y)/d(x)+2*y2)=0

3. Distribute and simplify the signs within the equation.

3*x2−3*x2d(y)/d(x)−6*x*y+4*x*yd(y)/d(x)+2*y2=0

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

4*x*yd(y)/d(x)−3*x2d(y)/d(x)=6*x*y−3*x2−2*y2

5. Factor out the common d(y)/d(x) term.

d(y)/d(x)*(4*x*y−3*x2)=6*x*y−3*x2−2*y2

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

d(y)/d(x)=(6*x*y−3*x2−2*y2)/(4*x*y−3*x2)

Final Answer

d(y)/d(x)=(6*x*y−3*x2−2*y2)/(4*x*y−3*x2)

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