Find dy/dx xy+y^2=1

Problem

x*y+y2=1

Solution

1. Differentiate both sides with respect to x treating y as a function of x

d()/d(x)*(x*y+y2)=d()/d(x)*1

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

(d(x)*y)/d(x)+d(y2)/d(x)=0

xd(y)/d(x)+yd(x)/d(x)+2*yd(y)/d(x)=0

3. Simplify the expression by replacing d(x)/d(x) with 1

xd(y)/d(x)+y+2*yd(y)/d(x)=0

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

xd(y)/d(x)+2*yd(y)/d(x)=−y

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

d(y)/d(x)*(x+2*y)=−y

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

d(y)/d(x)=(−y)/(x+2*y)

Final Answer

d(y)/d(x)=−y/(x+2*y)

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