Find dy/dx 2xy-y^2=1

Problem

2*x*y−y2=1

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)*(2*x*y−y2)=d()/d(x)*(1)

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

(d(2)*x*y)/d(x)−d(y2)/d(x)=0

2*y+2*xd(y)/d(x)−2*yd(y)/d(x)=0

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

2*xd(y)/d(x)−2*yd(y)/d(x)=−2*y

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

d(y)/d(x)*(2*x−2*y)=−2*y

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

d(y)/d(x)=(−2*y)/(2*x−2*y)

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

d(y)/d(x)=(−y)/(x−y)

7. Rewrite the expression by multiplying the numerator and denominator by −1 for a cleaner form.

d(y)/d(x)=y/(y−x)

Final Answer

d(y)/d(x)=y/(y−x)

Want more problems? Check here!