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

Problem

x2*y+3*x*y3−x=3

Solution

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

d()/d(x)*(x2*y+3*x*y3−x)=d()/d(x)*(3)

2. Apply the product rule to the terms x2*y and 3*x*y3

(x2d(y)/d(x)+y⋅2*x)+(3*x⋅3*y2d(y)/d(x)+3*y3⋅1)−1=0

3. Simplify the expression by multiplying the terms.

x2d(y)/d(x)+2*x*y+9*x*y2d(y)/d(x)+3*y3−1=0

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

x2d(y)/d(x)+9*x*y2d(y)/d(x)=1−2*x*y−3*y3

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

d(y)/d(x)*(x2+9*x*y2)=1−2*x*y−3*y3

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

d(y)/d(x)=(1−2*x*y−3*y3)/(x2+9*x*y2)

Final Answer

d(y)/d(x)=(1−2*x*y−3*y3)/(x2+9*x*y2)

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