Partial derivatives are an essential concept in calculus, particularly when dealing with functions of multiple variables. Let's break down the concept:

### 1. **Definition of Partial Derivatives**:
A **partial derivative** measures how a function of multiple variables changes as one of its variables is varied, while keeping the other variables constant. For a function \( f(x, y) \), the **partial derivative with respect to \(x\)** is denoted by:

\[
\frac{\partial f}{\partial x}
\]

and the **partial derivative with respect to \(y\)** is:

\[
\frac{\partial f}{\partial y}
\]

### 2. **Why Do We Use Partial Derivatives?**:
In many real-world applications, functions depend on more than one variable (e.g., temperature depending on both time and location). Partial derivatives allow us to study how changes in each individual variable affect the function while ignoring changes in the other variables.

### 3. **Notation**:
If you have a function \( f(x, y) \), its partial derivatives are written as:
- \( \frac{\partial f}{\partial x} \) or \( f_x \) (derivative with respect to \(x\)),
- \( \frac{\partial f}{\partial y} \) or \( f_y \) (derivative with respect to \(y\)).

### 4. **Computing Partial Derivatives**:
To compute a partial derivative, treat all other variables as constants. For example, if:

\[
f(x, y) = x^2 y + 3xy^2
\]

- The **partial derivative with respect to \(x\)** would be:

\[
\frac{\partial f}{\partial x} = 2xy + 3y^2
\]

- The **partial derivative with respect to \(y\)** would be:

\[
\frac{\partial f}{\partial y} = x^2 + 6xy
\]

### 5. **Higher-Order Partial Derivatives**:
Just like in single-variable calculus, you can take higher-order derivatives with partial derivatives:
- The **second-order partial derivatives** involve taking the derivative of a partial derivative. For example:

\[
\frac{\partial^2 f}{\partial x^2} \quad \text{or} \quad \frac{\partial^2 f}{\partial y^2}
\]

You can also compute **mixed partial derivatives**, such as:

\[
\frac{\partial^2 f}{\partial x \partial y}
\]

These measure how the function changes with respect to one variable, then the other.

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Would you like to explore any specific examples or details? Here are some related questions:

1. How do you interpret partial derivatives geometrically?
2. What is the chain rule for partial derivatives?
3. What are mixed partial derivatives, and when are they equal?
4. How are partial derivatives applied in optimization problems?
5. What is a gradient, and how is it related to partial derivatives?

**Tip**: When computing partial derivatives, always carefully treat other variables as constants.

Explore the fundamentals of partial derivatives, key concepts in multivariable calculus. Learn how to compute partial derivatives for functions of several variables, including higher-order and mixed partials, with real-world applications. Suitable for undergraduate calculus students.

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