We are given the equation:

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

Step 1: Differentiate both sides implicitly

Differentiate both sides with respect to $x$, remembering that $y$ is a function of $x$ (so we apply the chain rule to terms involving $y$).

$\frac{d}{dx} (3x^2 - y^2) = \frac{d}{dx} (4)$

Using differentiation rules:

• The derivative of $3x^2$ is $6x$.
• The derivative of $y^2$ is $2y \frac{dy}{dx}$ (using the chain rule).
• The derivative of $4$ is $0$.

Thus, we get:

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

Step 2: Solve for $\frac{dy}{dx}$

Rearrange the equation:

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

Divide by $-2y$:

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

Simplify:

$\frac{dy}{dx} = \frac{3x}{y}$

Final Answer:

$\frac{dy}{dx} = \frac{3x}{y}$

Would you like a deeper explanation on any step?

Here are some related questions to explore further:

1. How does the chain rule apply in implicit differentiation?
2. What happens if we differentiate both sides explicitly?
3. How can we find the second derivative $\frac{d^2y}{dx^2}$?
4. What are the points where $\frac{dy}{dx} = 0$?
5. How does implicit differentiation differ from explicit differentiation?

Tip: Always apply the chain rule when differentiating terms involving $y$, since $y$ is a function of $x$.