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

Problem

x2+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(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.

2*x+(xd(y)/d(x)+y⋅1)−2*yd(y)/d(x)=0

3. Group the terms containing d(y)/d(x) on one side of the equation and move all other terms to the opposite side.

xd(y)/d(x)−2*yd(y)/d(x)=−2*x−y

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

d(y)/d(x)*(x−2*y)=−2*x−y

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

d(y)/d(x)=(−2*x−y)/(x−2*y)

6. Simplify the expression by multiplying the numerator and denominator by −1 to adjust the signs.

d(y)/d(x)=(2*x+y)/(2*y−x)

Final Answer

d(y)/d(x)=(2*x+y)/(2*y−x)

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