Find dy/dx xy+2x+3y=1

Problem

x*y+2*x+3*y=1

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the product rule to the term x*y

(d(x)*y)/d(x)+(d(2)*x)/d(x)+(d(3)*y)/d(x)=d(1)/d(x)

2. Apply the product rule to x*y which gives xd(y)/d(x)+y(1)

xd(y)/d(x)+y+2+3d(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)+3d(y)/d(x)=−y−2

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

d(y)/d(x)*(x+3)=−y−2

5. Solve for dy/dx by dividing both sides by the expression (x+3)

d(y)/d(x)=(−y−2)/(x+3)

Final Answer

d(y)/d(x)=−(y+2)/(x+3)

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