Find dy/dx (x^2+y^2)^2=4x^2y

Problem

(x2+y2)2=4*x2*y

Solution

1. Differentiate both sides with respect to x using the chain rule on the left side and the product rule on the right side.

d(x2+y2)/d(x)=(d(4)*x2*y)/d(x)

2. Apply the chain rule to the left side, treating y as a function of x

2*(x2+y2)⋅(2*x+2*yd(y)/d(x))=(d(4)*x2*y)/d(x)

3. Apply the product rule to the right side.

2*(x2+y2)⋅(2*x+2*yd(y)/d(x))=8*x*y+4*x2d(y)/d(x)

4. Expand the terms on the left side to isolate the derivative terms.

4*x*(x2+y2)+4*y*(x2+y2)d(y)/d(x)=8*x*y+4*x2d(y)/d(x)

5. Group all terms containing d(y)/d(x) on one side and the remaining terms on the other side.

4*y*(x2+y2)d(y)/d(x)−4*x2d(y)/d(x)=8*x*y−4*x*(x2+y2)

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

d(y)/d(x)*(4*y*(x2+y2)−4*x2)=8*x*y−4*x*(x2+y2)

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

d(y)/d(x)=(8*x*y−4*x*(x2+y2))/(4*y*(x2+y2)−4*x2)

8. Simplify the fraction by dividing the numerator and denominator by 4.

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

9. Distribute and simplify the numerator and denominator further.

d(y)/d(x)=(2*x*y−x3−x*y2)/(y*x2+y3−x2)

Final Answer

d(y)/d(x)=(2*x*y−x3−x*y2)/(y*x2+y3−x2)

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