Find dy/dx x^2y^2-9x^2-4y^2=0

Problem

x2*y2−9*x2−4*y2=0

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)*y2)/d(x)−(d(9)*x2)/d(x)−(d(4)*y2)/d(x)=d(0)/d(x)

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

x2d(y2)/d(x)+y2d(x2)/d(x)−18*x−8*yd(y)/d(x)=0

3. Compute the derivatives using the chain rule for terms involving y

x2*(2*yd(y)/d(x))+y2*(2*x)−18*x−8*yd(y)/d(x)=0

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

2*x2*yd(y)/d(x)−8*yd(y)/d(x)=18*x−2*x*y2

5. Factor out the common d(y)/d(x) term.

d(y)/d(x)*(2*x2*y−8*y)=18*x−2*x*y2

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

d(y)/d(x)=(18*x−2*x*y2)/(2*x2*y−8*y)

7. Simplify the fraction by factoring out and canceling the common factor of 2

d(y)/d(x)=(2*(9*x−x*y2))/(2*(x2*y−4*y))

Final Answer

d(y)/d(x)=(9*x−x*y2)/(x2*y−4*y)

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