**Partial derivatives** are derivatives of multivariable functions with respect to one variable, while treating the other variables as constants. They play a crucial role in multivariable calculus and are used to understand how a function changes as one of its input variables changes.

### Key Points:
1. **Notation**:  
   The partial derivative of a function \( f(x_1, x_2, \dots, x_n) \) with respect to \( x_i \) is typically written as:
   \[
   \frac{\partial f}{\partial x_i}
   \]
   or \( f_{x_i} \).

2. **Procedure**:
   To compute a partial derivative of \( f(x_1, x_2, \dots, x_n) \) with respect to \( x_i \), treat all other variables \( x_j \) where \( j \neq i \) as constants, and differentiate as you would in single-variable calculus.

3. **Example**:
   Consider a function of two variables:
   \[
   f(x, y) = x^2 y + 3xy^2
   \]
   - The partial derivative with respect to \( x \) is:
   \[
   \frac{\partial f}{\partial x} = 2xy + 3y^2
   \]
   - The partial derivative with respect to \( y \) is:
   \[
   \frac{\partial f}{\partial y} = x^2 + 6xy
   \]

4. **Applications**:
   - **Optimization**: Partial derivatives help in finding local maxima, minima, and saddle points.
   - **Physics and Engineering**: Used in modeling phenomena involving multiple variables, such as heat and fluid flow.
   - **Gradient**: The vector of all partial derivatives of a function (denoted as \( \nabla f \)) provides the direction of the steepest ascent of the function.

5. **Higher-order Partial Derivatives**:  
   You can take the second (or higher) partial derivatives, such as:
   \[
   \frac{\partial^2 f}{\partial x^2}, \quad \frac{\partial^2 f}{\partial x \partial y}
   \]
   These are important in understanding curvature and concavity of surfaces.

Would you like more detailed examples or further clarification on any part of this?

### Follow-up Questions:
1. How do you compute the second-order partial derivatives?
2. What is the geometric interpretation of partial derivatives?
3. Can you explain the difference between partial and total derivatives?
4. How are partial derivatives used in Lagrange multipliers for optimization?
5. How do mixed partial derivatives relate to symmetry (Clairaut's theorem)?

**Tip**: When computing partial derivatives, always double-check which variables are being held constant. This helps prevent common errors when working with complex functions.

Explore partial derivatives, a key concept in multivariable calculus. Learn how to compute partial derivatives, including examples, formulas, and applications in optimization and physics. Suitable for undergraduate students studying advanced calculus topics.

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