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Find derivative by implicit differentiation 

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 \(y\).

Find derivative by implicit differentiation

Learn how to find the derivative of the equation x^2 - 6xy + y^2 = 6 using implicit differentiation. This solution involves applying the product rule, chain rule, and implicit differentiation techniques. Suitable for students in Grades 11-12.

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