Find dy/dx x^3+y^3=9xy

Problem

x^3+y^3=9*x*y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule to the y^3 term.

d(x^3)/d(x)+d(y^3)/d(x)=(d(9)*x*y)/d(x)

2. Apply the power rule to the left side and the product rule to the right side.

3*x^2+3*y^2d(y)/d(x)=9*y+9*xd(y)/d(x)

3. Group the terms containing d(y)/d(x) on one side of the equation and the remaining terms on the other side.

3*y^2d(y)/d(x)−9*xd(y)/d(x)=9*y−3*x^2

4. Factor out the common d(y)/d(x) term from the left side.

d(y)/d(x)*(3*y^2−9*x)=9*y−3*x^2

5. Isolate the derivative by dividing both sides by (3*y^2−9*x)

d(y)/d(x)=(9*y−3*x^2)/(3*y^2−9*x)

6. Simplify the fraction by dividing the numerator and the denominator by their greatest common factor, 3.

d(y)/d(x)=(3*y−x^2)/(y^2−3*x)

Final Answer

d(y)/d(x)=(3*y−x^2)/(y^2−3*x)

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