Find $dy/dx$ for the equation below using implicit differentiation.
$5x^2+y^3=12$

• A. $dy/dx=-\frac{10x}{3y^2}$
• B. $dy/dx=\frac{10x}{3y^2}$
• C. $dy/dx=\left(12-5x^2\right)^{\left(\frac13\right)}$
• D. $dy/dx=-\left(12-5x^2\right)^{\left(\frac13\right)}$

Answer: (A)

Verified step by step guidance

1. Step 1: Start with the given equation: 5x^2 + y^3 = 12. The goal is to find dy/dx using implicit differentiation.
2. Step 2: Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, apply the chain rule. The derivative of 5x^2 is 10x, and the derivative of y^3 is 3y^2(dy/dx).
3. Step 3: After differentiating, the equation becomes: 10x + 3y^2(dy/dx) = 0.
4. Step 4: Solve for dy/dx by isolating it. Subtract 10x from both sides to get: 3y^2(dy/dx) = -10x.
5. Step 5: Divide both sides by 3y^2 to isolate dy/dx, resulting in: dy/dx = -10x / (3y^2).