Find dy/dx x^2+4xy-y^3=6

Problem

x2+4*x*y−y3=6

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(x2)/d(x)+(d(4)*x*y)/d(x)−d(y3)/d(x)=d(6)/d(x)

2. Apply the product rule to the term 4*x*y and the power rule with the chain rule to the other terms.

2*x+(4*y+4*xd(y)/d(x))−3*y2d(y)/d(x)=0

3. Group all terms containing d(y)/d(x) on one side of the equation and move all other terms to the opposite side.

4*xd(y)/d(x)−3*y2d(y)/d(x)=−2*x−4*y

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

d(y)/d(x)*(4*x−3*y2)=−2*x−4*y

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

d(y)/d(x)=(−2*x−4*y)/(4*x−3*y2)

Final Answer

d(y)/d(x)=(2*x+4*y)/(3*y2−4*x)

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