Find dy/dx y^2=(x^2)/(xy-2500)

Problem

y2=(x2)/(x*y−2500)

Solution

1. Clear the fraction by multiplying both sides of the equation by the denominator x*y−2500

y2*(x*y−2500)=x2

2. Distribute the y2 term on the left side.

x*y3−2500*y2=x2

3. Differentiate implicitly with respect to x applying the product rule to x*y3 and the chain rule to terms involving y

(d(x)*y3)/d(x)−(d(2500)*y2)/d(x)=d(x2)/d(x)

y3+3*x*y2d(y)/d(x)−5000*yd(y)/d(x)=2*x

4. Isolate the terms containing d(y)/d(x) on one side of the equation.

3*x*y2d(y)/d(x)−5000*yd(y)/d(x)=2*x−y3

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

d(y)/d(x)*(3*x*y2−5000*y)=2*x−y3

6. Solve for the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(2*x−y3)/(3*x*y2−5000*y)

Final Answer

d(y)/d(x)=(2*x−y3)/(3*x*y2−5000*y)

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