Find dy/dx 24x^3y^2-8y^3=-11

Problem

24*x3*y2−8*y3=−11

Solution

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

d()/d(x)*(24*x3*y2−8*y3)=d()/d(x)*(−11)

2. Apply the product rule to the first term 24*x3*y2 and the chain rule to the second term 8*y3

24*(3*x2*y2+x3⋅2*yd(y)/d(x))−24*y2d(y)/d(x)=0

3. Distribute the constant and simplify the terms.

72*x2*y2+48*x3*yd(y)/d(x)−24*y2d(y)/d(x)=0

4. Isolate the terms containing d(y)/d(x) on one side of the equation.

48*x3*yd(y)/d(x)−24*y2d(y)/d(x)=−72*x2*y2

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

d(y)/d(x)*(48*x3*y−24*y2)=−72*x2*y2

6. Solve for dy/dx by dividing both sides by the factored expression.

d(y)/d(x)=(−72*x2*y2)/(48*x3*y−24*y2)

7. Simplify the fraction by dividing the numerator and denominator by the greatest common factor 24*y

d(y)/d(x)=(−3*x2*y)/(2*x3−y)

Final Answer

d(y)/d(x)=(3*x2*y)/(y−2*x3)

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