Find dy/dx x^3y^3-y=x

Problem

x3*y3−y=x

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)*(x3*y3−y)=d(x)/d(x)

2. Apply the product rule to the term x3*y3 and the power rule to the other terms.

(d(x3)*y3)/d(x)−d(y)/d(x)=1

(3*x2)*(y3)+(x3)*(3*y2d(y)/d(x))−d(y)/d(x)=1

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

3*x3*y2d(y)/d(x)−d(y)/d(x)=1−3*x2*y3

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

d(y)/d(x)*(3*x3*y2−1)=1−3*x2*y3

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

d(y)/d(x)=(1−3*x2*y3)/(3*x3*y2−1)

Final Answer

d(y)/d(x)=(1−3*x2*y3)/(3*x3*y2−1)

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