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### **Exercise 1 (Part 1):**
We are given the function:
\[
f(x, y) = \frac{x - 2y}{x^2 + 3y^2 + 1}
\]

We need to:
1. Prove that \( f \) is differentiable at all points.
2. Compute \( df(2, 0) \).
3. Write the first-order Taylor expansion (limited development) of \( f \) near the point \( (2, 0) \).

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#### **Step 1: Prove differentiability at all points**

To prove \( f(x, y) \) is differentiable at all points, we need to:
- Ensure \( f \) is continuous.
- Check that the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist and are continuous.

The function \( f(x, y) \) is a rational function, so:
- The denominator \( x^2 + 3y^2 + 1 \) is strictly positive for all real \( x \) and \( y \). Hence, \( f(x, y) \) is continuous everywhere.
- The partial derivatives exist as long as the denominator is non-zero, which holds true everywhere. 

The partial derivatives are given by the quotient rule.

\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( \frac{x - 2y}{x^2 + 3y^2 + 1} \right)
= \frac{(1)(x^2 + 3y^2 + 1) - (x - 2y)(2x)}{(x^2 + 3y^2 + 1)^2}
\]

Simplifying:
\[
\frac{\partial f}{\partial x} = \frac{x^2 + 3y^2 + 1 - 2x^2 + 4xy}{(x^2 + 3y^2 + 1)^2}
= \frac{-x^2 + 4xy + 3y^2 + 1}{(x^2 + 3y^2 + 1)^2}
\]

Similarly, for \( \frac{\partial f}{\partial y} \):
\[
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x - 2y}{x^2 + 3y^2 + 1} \right)
= \frac{(-2)(x^2 + 3y^2 + 1) - (x - 2y)(6y)}{(x^2 + 3y^2 + 1)^2}
\]

Simplifying:
\[
\frac{\partial f}{\partial y} = \frac{-2x^2 - 6y^2 - 2 + 6xy - 12y^2}{(x^2 + 3y^2 + 1)^2}
= \frac{-2x^2 + 6xy - 18y^2 - 2}{(x^2 + 3y^2 + 1)^2}
\]

Both partial derivatives are continuous because the denominator is non-zero everywhere. Thus, \( f(x, y) \) is differentiable everywhere.

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#### **Step 2: Compute \( df(2, 0)

Show that the function f(x, y) = (x - 2y) / (x^2 + 3y^2 + 1) is differentiable at all points. Compute df(2, 0) and write the first-order Taylor expansion of f near (2, 0). Additionally, address similar differentiability for another function and verify differentiability for specific cases.

Explore the differentiability of the function f(x, y) = (x - 2y) / (x^2 + 3y^2 + 1). Learn how to compute df(2, 0), find the first-order Taylor expansion, and analyze differentiability for related cases using multivariable calculus concepts.

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