Let's solve the problem step by step.

Problem:

We are given the curve equation:

$2y^2 - 4y = x$

We need to find the value of $\frac{dy}{dx}$ at the point $(3, 6)$.

Step 1: Differentiate implicitly

To find $\frac{dy}{dx}$, we differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (implicit differentiation):

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

Differentiating term by term:

1. For $2y^2$, use the chain rule: $\frac{d}{dx}(2y^2) = 2 \cdot 2y \cdot \frac{dy}{dx} = 4y \frac{dy}{dx}.$
2. For $-4y$, use the chain rule: $\frac{d}{dx}(-4y) = -4 \cdot \frac{dy}{dx}.$
3. For $x$: $\frac{d}{dx}(x) = 1.$

Thus, the differentiated equation becomes: $4y \frac{dy}{dx} - 4 \frac{dy}{dx} = 1.$

Step 2: Factorize and solve for $\frac{dy}{dx}$

Factor $\frac{dy}{dx}$ out: $\frac{dy}{dx} \left( 4y - 4 \right) = 1.$

Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{4y - 4}.$

Step 3: Substitute the given point $(3, 6)$

At the point $(x, y) = (3, 6)$, substitute $y = 6$ into the expression for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{1}{4(6) - 4}.$

Simplify the denominator: $\frac{dy}{dx} = \frac{1}{24 - 4} = \frac{1}{20}.$

Final Answer:

The value of $\frac{dy}{dx}$ at the point $(3, 6)$ is: $\boxed{\frac{1}{20}}.$

Would you like me to elaborate on any step, or do you have additional questions?

Follow-Up Questions:

1. What does implicit differentiation involve, and why is it used here?
2. How does the chain rule apply when differentiating $y$ with respect to $x$?
3. Can you use explicit differentiation here instead of implicit differentiation?
4. How would you interpret $\frac{dy}{dx}$ geometrically at the given point?
5. What happens to the value of $\frac{dy}{dx}$ as $y$ approaches infinity?

Tip:

Always substitute given points into the differentiated equation only after solving for the derivative explicitly. This minimizes errors in calculations.