Find dy/dx 2(x^2+y^2)^2=25(x^2-y^2)

Problem

2*(x2+y2)2=25*(x2−y2)

Solution

1. Differentiate both sides with respect to x using the chain rule and the power rule.

2⋅2*(x2+y2)(2−1)⋅d(x2+y2)/d(x)=25⋅d(x2−y2)/d(x)

2. Apply implicit differentiation to the terms involving y treating y as a function of x

4*(x2+y2)*(2*x+2*yd(y)/d(x))=25*(2*x−2*yd(y)/d(x))

3. Expand the expressions on both sides to isolate the terms containing d(y)/d(x)

8*x*(x2+y2)+8*y*(x2+y2)d(y)/d(x)=50*x−50*yd(y)/d(x)

4. Group the terms with d(y)/d(x) on the left side and the remaining terms on the right side.

8*y*(x2+y2)d(y)/d(x)+50*yd(y)/d(x)=50*x−8*x*(x2+y2)

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

d(y)/d(x)*(8*y*(x2+y2)+50*y)=50*x−8*x*(x2+y2)

6. Solve for d(y)/d(x) by dividing both sides by the factor in parentheses.

d(y)/d(x)=(50*x−8*x*(x2+y2))/(8*y*(x2+y2)+50*y)

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

d(y)/d(x)=(25*x−4*x*(x2+y2))/(4*y*(x2+y2)+25*y)

Final Answer

d(y)/d(x)=(25*x−4*x*(x2+y2))/(4*y*(x2+y2)+25*y)

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