Find dy/dx xy^2-x^3y=6

Problem

x*y2−x3*y=6

Solution

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

d()/d(x)*(x*y2−x3*y)=d(6)/d(x)

2. Apply the product rule to the first term x*y2

(d(x)*y2)/d(x)=y2d(x)/d(x)+xd(y2)/d(x)

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

3. Apply the product rule to the second term −x3*y

d(−x3*y)/d(x)=−(yd(x3)/d(x)+x3d(y)/d(x))

−(3*x2*y+x3d(y)/d(x))

4. Combine the results and set the derivative of the constant on the right side to zero.

y2+2*x*yd(y)/d(x)−3*x2*y−x3d(y)/d(x)=0

5. Isolate the terms containing d(y)/d(x) on one side of the equation.

2*x*yd(y)/d(x)−x3d(y)/d(x)=3*x2*y−y2

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

d(y)/d(x)*(2*x*y−x3)=3*x2*y−y2

7. Solve for d(y)/d(x) by dividing both sides by the factored expression.

d(y)/d(x)=(3*x2*y−y2)/(2*x*y−x3)

Final Answer

d(y)/d(x)=(3*x2*y−y2)/(2*x*y−x3)

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