Find dy/dx x^2y^2-xy=8

Problem

x2*y2−x*y=8

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule where necessary.

d(x2*y2−x*y)/d(x)=d(8)/d(x)

2. Apply the product rule to the first term x2*y2 and the second term x*y

(d(x2)*y2)/d(x)−(d(x)*y)/d(x)=0

(x2⋅2*yd(y)/d(x)+y2⋅2*x)−(xd(y)/d(x)+y⋅1)=0

3. Distribute and simplify the expression to remove parentheses.

2*x2*yd(y)/d(x)+2*x*y2−xd(y)/d(x)−y=0

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

2*x2*yd(y)/d(x)−xd(y)/d(x)=y−2*x*y2

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

d(y)/d(x)*(2*x2*y−x)=y−2*x*y2

6. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(y−2*x*y2)/(2*x2*y−x)

Final Answer

d(y)/d(x)=(y−2*x*y2)/(2*x2*y−x)

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