Find dy/dx x^2y^3-2xy^2=5

Problem

x^2*y^3−2*x*y^2=5

Solution

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

d()/d(x)*(x^2*y^3−2*x*y^2)=d(5)/d(x)

2. Apply the product rule to the first term x^2*y^3 which gives u^′*v+u*v^′

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

3. Apply the product rule to the second term −2*x*y^2

−2*(d(x)/d(x)*y^2+xd(y^2)/d(x))=−2*y^2−4*x*yd(y)/d(x)

4. Combine the results and set the derivative of the constant on the right side to zero.

2*x*y^3+3*x^2*y^2d(y)/d(x)−2*y^2−4*x*yd(y)/d(x)=0

5. Group the terms containing d(y)/d(x) on one side and move all other terms to the other side.

3*x^2*y^2d(y)/d(x)−4*x*yd(y)/d(x)=2*y^2−2*x*y^3

6. Factor out d(y)/d(x) from the left side of the equation.

d(y)/d(x)*(3*x^2*y^2−4*x*y)=2*y^2−2*x*y^3

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

d(y)/d(x)=(2*y^2−2*x*y^3)/(3*x^2*y^2−4*x*y)

8. Simplify the fraction by factoring out y from the numerator and x*y from the denominator.

d(y)/d(x)=(2*y*(y−x*y^2))/(x*y*(3*x*y−4))

d(y)/d(x)=(2*y−2*x*y^2)/(3*x^2*y−4*x)

Final Answer

d(y)/d(x)=(2*y^2−2*x*y^3)/(3*x^2*y^2−4*x*y)

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