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

Problem

x2−4*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(x2)/d(x)−(d(4)*x*y)/d(x)+d(y2)/d(x)=d(4)/d(x)

2. Apply the product rule to the term 4*x*y and the chain rule to the term y2

2*x−(4*y+4*xd(y)/d(x))+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.

2*x−4*y−4*xd(y)/d(x)+2*yd(y)/d(x)=0

−4*xd(y)/d(x)+2*yd(y)/d(x)=4*y−2*x

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

d(y)/d(x)*(2*y−4*x)=4*y−2*x

5. Solve for d(y)/d(x) by dividing both sides by (2*y−4*x)

d(y)/d(x)=(4*y−2*x)/(2*y−4*x)

6. Simplify the fraction by dividing the numerator and denominator by 2

d(y)/d(x)=(2*y−x)/(y−2*x)

Final Answer

d(y)/d(x)=(2*y−x)/(y−2*x)

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