Find $$ d y / d x $$ for $$ 3 x^{2}-y^{2}=4 $$ Type answer $$ y^{\prime}= $$ (numerator)/(denominator) A

Question

UpStudy AI · Accepted Answer

Differentiate the given equation with respect to $$ x $$:

$$
3x^2 - y^2 = 4
$$

Differentiate term-by-term:

1. For $$ 3x^2 $$, the derivative is:
   $$
   \frac{d}{dx}(3x^2) = 6x
   $$

2. For $$ -y^2 $$, note that $$ y $$ is a function of $$ x $$. Using the chain rule:
   $$
   \frac{d}{dx}(-y^2) = -2y \frac{dy}{dx}
   $$

3. The derivative of the constant $$ 4 $$ is:
   $$
   \frac{d}{dx}(4) = 0
   $$

Putting it together:

$$
6x - 2y\frac{dy}{dx} = 0
$$

Solve for $$ \frac{dy}{dx} $$:

$$
-2y\frac{dy}{dx} = -6x \quad \Rightarrow \quad \frac{dy}{dx} = \frac{-6x}{-2y} = \frac{6x}{2y} = \frac{3x}{y}
$$

Thus, the derivative is:

$$
y^{\prime} = \frac{3x}{y}
$$
$$ y^{\prime} = \frac{3x}{y} $$

Find \( d y / d x \) for \( 3 x^{2}-y^{2}=4 \) Type answer \( y^{\prime}= \) (numerator)/(denominator) A