Find dy/dx x^2y^2-2x=3

Problem

x2*y2−2*x=3

Solution

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

d()/d(x)*(x2*y2−2*x)=d()/d(x)*3

2. Apply the product rule to the term x2*y2 and the power rule to the other terms.

(d(x2)*y2)/d(x)−(d(2)*x)/d(x)=0

3. Expand the derivative of the product, remembering to multiply by d(y)/d(x) when differentiating y terms.

x2d(y2)/d(x)+y2d(x2)/d(x)−2=0

x2*(2*yd(y)/d(x))+y2*(2*x)−2=0

4. Simplify the expression to group the terms.

2*x2*yd(y)/d(x)+2*x*y2−2=0

5. Isolate the term containing d(y)/d(x) by moving all other terms to the right side of the equation.

2*x2*yd(y)/d(x)=2−2*x*y2

6. Solve for dy/dx by dividing both sides by the coefficient 2*x2*y

d(y)/d(x)=(2−2*x*y2)/(2*x2*y)

7. Factor and simplify the fraction by dividing the numerator and denominator by 2.

d(y)/d(x)=(2*(1−x*y2))/(2*x2*y)

d(y)/d(x)=(1−x*y2)/(x2*y)

Final Answer

d(y)/d(x)=(1−x*y2)/(x2*y)

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