Find dy/dx 6(x^2+y^2)^2=169(x^2-y^2)

Problem

6*(x2+y2)2=169*(x2−y2)

Solution

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

d()/d(x)*6*(x2+y2)2=d()/d(x)*169*(x2−y2)

2. Apply the chain rule to the left side and the power rule to the right side, remembering that y is a function of x

12*(x2+y2)⋅(2*x+2*yd(y)/d(x))=169*(2*x−2*yd(y)/d(x))

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

24*x*(x2+y2)+24*y*(x2+y2)d(y)/d(x)=338*x−338*yd(y)/d(x)

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

24*y*(x2+y2)d(y)/d(x)+338*yd(y)/d(x)=338*x−24*x*(x2+y2)

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

d(y)/d(x)*(24*y*(x2+y2)+338*y)=338*x−24*x*(x2+y2)

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

d(y)/d(x)=(338*x−24*x*(x2+y2))/(24*y*(x2+y2)+338*y)

7. Simplify the expression by factoring out 2*x from the numerator and 2*y from the denominator, then canceling the common factor of 2

d(y)/d(x)=(2*x*(169−12*(x2+y2)))/(2*y*(12*(x2+y2)+169))

Final Answer

d(y)/d(x)=(x*(169−12*x2−12*y2))/(y*(169+12*x2+12*y2))

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