4. Applications of Derivatives

Implicit Differentiation

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Multiple Choice

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

$dy/dx=-\frac{10x}{3y^2}$

$dy/dx=\frac{10x}{3y^2}$

$dy/dx=\left(12-5x^2\right)^{\left(\frac13\right)}$

$dy/dx=-\left(12-5x^2\right)^{\left(\frac13\right)}$

Verified step by step guidance

Start with the given equation: 5x^2 + y^3 = 12.

Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so use the chain rule for terms involving y.

The derivative of 5x^2 with respect to x is 10x.

For the term y^3, apply the chain rule: the derivative is 3y^2 * (dy/dx).

Set the derivatives equal to the derivative of the constant 12, which is 0, and solve for dy/dx: 10x + 3y^2 * (dy/dx) = 0. Rearrange to find dy/dx.

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Find dy/dxdy/dxdy/dx for the equation below using implicit differentiation.5x2+y3=125x^2+y^3=125x2+y3=12