find dy/dx by implicit differentiation x^2-4xy+y^2=4

To find \(\frac{dy}{dx}\) by implicit differentiation of the equation \(x^2 - 4xy + y^2 = 4\), follow these steps:

1. **Differentiate both sides with respect to \(x\)**, keeping in mind that \(y\) is a function of \(x\), so you must apply the chain rule when differentiating terms involving \(y\).

\[
\frac{d}{dx}(x^2 - 4xy + y^2) = \frac{d}{dx}(4)
\]

2. **Differentiate each term**:

- \(\frac{d}{dx}(x^2) = 2x\)
- \(\frac{d}{dx}(-4xy)\): Use the product rule, \(\frac{d}{dx}(uv) = u'v + uv'\), where \(u = -4x\) and \(v = y\).
\[
\frac{d}{dx}(-4xy) = (-4)(y) + (-4x)\left(\frac{dy}{dx}\right) = -4y - 4x\frac{dy}{dx}
\]
- \(\frac{d}{dx}(y^2)\): Using the chain rule, \(\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}\).

The right side of the equation is a constant, so \(\frac{d}{dx}(4) = 0\).

Now we have:
\[
2x - 4y - 4x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0
\]

3. **Solve for \(\frac{dy}{dx}\)**:

Group the terms involving \(\frac{dy}{dx}\) on one side:
\[
- 4x\frac{dy}{dx} + 2y\frac{dy}{dx} = -2x + 4y
\]

Factor out \(\frac{dy}{dx}\):
\[
\frac{dy}{dx}(-4x + 2y) = -2x + 4y
\]

Finally, solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-2x + 4y}{-4x + 2y}
\]

Simplifying the expression:
\[
\frac{dy}{dx} = \frac{2(y - x)}{2y - 4x} = \frac{y - x}{y - 2x}
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = \frac{y - x}{y - 2x}
\]

Would you like more details or further clarification on any part of this solution?

Here are five related questions to expand your understanding:
1. How would you apply implicit differentiation to more complex functions?
2. What happens when we differentiate trigonometric implicit functions?
3. How can you verify that the expression for \(\frac{dy}{dx}\) is correct by substituting specific values for \(x\) and \(y\)?
4. How does implicit differentiation handle higher-order derivatives?
5. Can implicit differentiation be used to find tangent lines to curves at specific points?

**Tip:** When performing implicit differentiation, remember to apply the chain rule carefully whenever differentiating a function involving \(y\), since \(y\) depends on \(x\).

Learn how to find dy/dx using implicit differentiation for the equation x^2 - 4xy + y^2 = 4. This solution covers key calculus concepts like the chain rule and product rule, suitable for Grade 12 or undergraduate calculus students.

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