Find dy/dx x^2y^2=100

Problem

x^2*y^2=100

Solution

1. Differentiate both sides of the equation with respect to x using the chain rule and the product rule.

(d(x^2)*y^2)/d(x)=d(100)/d(x)

2. Apply the product rule to the left side, where the derivative of u⋅v is u^′*v+u*v^′

y^2d(x^2)/d(x)+x^2d(y^2)/d(x)=0

3. Compute the derivatives, remembering to use the chain rule for the term involving y

y^2*(2*x)+x^2*(2*yd(y)/d(x))=0

4. Simplify the expression to prepare for solving for d(y)/d(x)

2*x*y^2+2*x^2*yd(y)/d(x)=0

5. Isolate the term containing d(y)/d(x) by subtracting 2*x*y^2 from both sides.

2*x^2*yd(y)/d(x)=−2*x*y^2

6. Solve for dy/dx by dividing both sides by 2*x^2*y

d(y)/d(x)=(−2*x*y^2)/(2*x^2*y)

7. Simplify the fraction by canceling common factors of 2 x and y

d(y)/d(x)=−y/x

Final Answer

d(y)/d(x)=−y/x

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