Find dy/dx 3(x^2+y^2)^2=100(x^2-y^2)

Problem

3*(x2+y2)2=100*(x2−y2)

Solution

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

d()/d(x)*3*(x2+y2)2=d()/d(x)*100*(x2−y2)

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

6*(x2+y2)⋅(2*x+2*yd(y)/d(x))=100*(2*x−2*yd(y)/d(x))

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

12*x*(x2+y2)+12*y*(x2+y2)d(y)/d(x)=200*x−200*yd(y)/d(x)

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

12*y*(x2+y2)d(y)/d(x)+200*yd(y)/d(x)=200*x−12*x*(x2+y2)

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

d(y)/d(x)*(12*y*(x2+y2)+200*y)=200*x−12*x*(x2+y2)

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

d(y)/d(x)=(200*x−12*x*(x2+y2))/(12*y*(x2+y2)+200*y)

7. Simplify the fraction by factoring out and canceling common factors (in this case, 4).

d(y)/d(x)=(4*(50*x−3*x*(x2+y2)))/(4*(3*y*(x2+y2)+50*y))

Final Answer

d(y)/d(x)=(50*x−3*x*(x2+y2))/(3*y*(x2+y2)+50*y)

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