To find the derivative of the implicit function \( xy + x = 3y \) with respect to \( x \), we'll use implicit differentiation. Here’s the step-by-step process:

Given the equation:
\[
xy + x = 3y
\]

1. Differentiate both sides of the equation with respect to \( x \):

\[
\frac{d}{dx}(xy) + \frac{d}{dx}(x) = \frac{d}{dx}(3y)
\]

2. Apply the product rule to the term \( xy \):

\[
\frac{d}{dx}(xy) = y \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(y) = y + x \frac{dy}{dx}
\]

3. The derivative of \( x \) with respect to \( x \) is 1, and the derivative of \( 3y \) with respect to \( x \) is \( 3 \frac{dy}{dx} \):

\[
y + x \frac{dy}{dx} + 1 = 3 \frac{dy}{dx}
\]

4. Now, isolate \( \frac{dy}{dx} \):

\[
x \frac{dy}{dx} - 3 \frac{dy}{dx} = -y - 1
\]

\[
\frac{dy}{dx}(x - 3) = -y - 1
\]

5. Finally, solve for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{-y - 1}{x - 3}
\]

So, the derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = \frac{-y - 1}{x - 3}
\]

Would you like more details on any specific step or have any questions?

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Here are five related questions for further exploration:

1. How would you find the second derivative of \( y \) with respect to \( x \) for the same equation?
2. What is the geometric interpretation of the derivative \( \frac{dy}{dx} \) in this context?
3. How does implicit differentiation differ from explicit differentiation?
4. Can you solve for \( y \) explicitly in terms of \( x \) from the given equation?
5. How would you find the points where the tangent line to the curve is horizontal?

**Tip:** When dealing with implicit differentiation, remember to apply the product rule carefully when differentiating terms like \( xy \).

Learn how to find the derivative of xy + x = 3y using implicit differentiation with a step-by-step solution. Understand the application of implicit differentiation in calculus. Suitable for advanced high school students and college calculus students.

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