Define symbolic variables

A partial derivative measures how the output changes with respect to one input variable while holding all other variables constant. Collectively, all first-order partial derivatives form the gradient vector.

I explicitly identify which variable is being differentiated—and which variables are temporarily treated as constants. Confusing these roles leads directly to incorrect gradient calculations.

In the previous section, we explored definite integrals and their applications—learning how to compute basic definite integrals and how they apply to real-world problems. In this section, we shift our focus to multivariable calculus, centering on multivariable functions and their partial derivatives. This foundation will prepare us for the upcoming discussion of multiple integrals.

Multivariable Functions

A multivariable function is one whose output depends on two or more independent variables. For a function $f(x, y)$, where $x$ and $y$ are independent variables, we can interpret it as a function defined over the plane; graphically, such a function corresponds to a surface in three-dimensional space.

Example

Consider the multivariable function $f(x, y) = x^2 + y^2$. Here, the function value equals the squared Euclidean distance from point $(x, y)$ to the origin $(0, 0)$. Geometrically, this function describes a symmetric, upward-opening paraboloid (a “bowl-shaped” surface), symmetric about both coordinate axes.

Partial Derivatives

A partial derivative of a multivariable function is its derivative with respect to one independent variable, while all other independent variables are held fixed. For $f(x, y)$, we compute its partial derivatives with respect to $x$ and $y$, denoted $f_x(x, y)$ and $f_y(x, y)$, respectively.

Definition

The formal definitions are:

• Partial derivative with respect to $x$:

  $$f_x(x, y) = \frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}$$

• Partial derivative with respect to $y$:

  $$f_y(x, y) = \frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}$$

Example Calculation

Let’s compute the partial derivatives of $f(x, y) = x^2 + y^2$:

1. Compute $f_x(x, y)$:

   $$f_x(x, y) = \frac{\partial}{\partial x}(x^2 + y^2) = 2x$$

   This means that, with $y$ held constant, the instantaneous rate of change of $f$ with respect to $x$ is $2x$.

2. Compute $f_y(x, y)$:

   $$f_y(x, y) = \frac{\partial}{\partial y}(x^2 + y^2) = 2y$$

   This means that, with $x$ held constant, the instantaneous rate of change of $f$ with respect to $y$ is $2y$.

Physical Interpretation

Partial derivatives play a vital role in applied contexts. For instance, in heat conduction problems, temperature distribution is modeled by a multivariable function, and partial derivatives quantify how temperature changes along specific spatial directions.

Case Study: Gradient and Extrema

Beyond measuring local rates of change, partial derivatives help locate extrema (maxima and minima) of multivariable functions—via the gradient.

When learning multivariable functions and partial derivatives, always begin by clearly stating:

• Which variables are independent,
• Which variables are held constant,
• With respect to which variable you’re differentiating,
• Then interpret the result as the instantaneous rate of change in that direction.

The gradient vector is defined as

$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right).$$

Its direction points toward the steepest ascent of $f$, and its magnitude equals the rate of increase in that direction.

Example: Classifying Critical Points

Consider $f(x, y) = -x^2 - y^2 + 4$. To find and classify extrema:

1. Compute partial derivatives:

   $$f_x(x, y) = -2x,\quad f_y(x, y) = -2y$$

2. Set both partial derivatives to zero:

   $$-2x = 0 \Rightarrow x = 0,\quad -2y = 0 \Rightarrow y = 0$$

   So $(0, 0)$ is a critical point.

3. Use the second derivative test (e.g., via the Hessian matrix) to determine the nature of this point.

The analysis confirms that $(0, 0)$ is a local maximum, and the maximum value of $f$ is $4$.

Python Implementation

We can compute partial derivatives numerically or symbolically using Python—for example, with the sympy library:

Read “Multivariable Functions and Partial Derivatives” through the lens of Scenario → Concept → Action → Result. First align these four elements, then revisit parameters, code, or procedural steps in the main text.

import sympy as sp

# Define symbolic variables
x, y = sp.symbols('x y')

# Define the function
f = x**2 + y**2

# Compute partial derivatives
f_x = sp.diff(f, x)
f_y = sp.diff(f, y)

print(f'Partial derivative w.r.t. x: {f_x}')
print(f'Partial derivative w.r.t. y: {f_y}')

Running this code yields the symbolic partial derivatives, enabling further analytical or numerical exploration.

Having read this section, consolidate “Multivariable Functions and Partial Derivatives” into a structured recap table: clarify the central narrative first, then validate understanding using a small, concrete task.

After finishing “Multivariable Functions and Partial Derivatives”, pick a simple example and walk through the full workflow end-to-end. Then assess which steps you can now execute independently.

Summary

In this section, we introduced multivariable functions and partial derivatives. We learned how to define and compute partial derivatives, and explored their significance in modeling real-world phenomena. Mastering these concepts lays essential groundwork for the next section: computing and applying multiple integrals. There, we’ll delve deeper into double and triple integrals—their evaluation techniques and diverse applications.