Find dy/dx y=x^2y^3+x^3y^2

Problem

y=x2*y3+x3*y2

Solution

1. Identify the equation as an implicit function where y is a function of x requiring implicit differentiation.

2. Differentiate both sides of the equation with respect to x applying the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) to both terms on the right side.

3. Apply the chain rule to terms involving y such that d(yn)/d(x)=n*y(n−1)d(y)/d(x)

d(y)/d(x)=(d(x2)*y3)/d(x)+(d(x3)*y2)/d(x)

d(y)/d(x)=(x2⋅3*y2d(y)/d(x)+y3⋅2*x)+(x3⋅2*yd(y)/d(x)+y2⋅3*x2)

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

d(y)/d(x)−3*x2*y2d(y)/d(x)−2*x3*yd(y)/d(x)=2*x*y3+3*x2*y2

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

d(y)/d(x)*(1−3*x2*y2−2*x3*y)=2*x*y3+3*x2*y2

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

d(y)/d(x)=(2*x*y3+3*x2*y2)/(1−3*x2*y2−2*x3*y)

Final Answer

d(y)/d(x)=(2*x*y3+3*x2*y2)/(1−3*x2*y2−2*x3*y)

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