The equation shown is:

$x^2 - 6xy + y^2 = 6$

We will find the derivative $\frac{dy}{dx}$ using implicit differentiation.

Step-by-step process:

1. Differentiate both sides of the equation with respect to $x$. Remember, $y$ is a function of $x$, so when differentiating terms involving $y$, apply the chain rule.

$\frac{d}{dx}(x^2) - \frac{d}{dx}(6xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(6)$

2. Apply differentiation:
   • For $x^2$, the derivative is $2x$.
   • For $6xy$, use the product rule: $\frac{d}{dx}(6xy) = 6\left(\frac{d}{dx}(x) \cdot y + x \cdot \frac{dy}{dx}\right) = 6(y + x \frac{dy}{dx})$
   • For $y^2$, apply the chain rule: $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$
   • The derivative of the constant 6 is 0.

So, the differentiated equation becomes:

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

3. Simplify the equation: $2x - 6y - 6x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$

4. Collect all terms involving $\frac{dy}{dx}$ on one side: $-6x \frac{dy}{dx} + 2y \frac{dy}{dx} = 6y - 2x$

5. Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(-6x + 2y) = 6y - 2x$

6. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{6y - 2x}{2y - 6x}$

Final answer:

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

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen understanding:

1. How do you apply the product rule in implicit differentiation?
2. What is the significance of using the chain rule for $y$ terms in implicit differentiation?
3. Can implicit differentiation be used to find higher-order derivatives?
4. How would you handle implicit differentiation if the equation had more complicated terms like trigonometric functions?
5. Why do we collect terms involving $\frac{dy}{dx}$ on one side of the equation?

Tip: In implicit differentiation, always remember to treat $y$ as a function of $x$ and apply the chain rule when differentiating terms involving $y$.