Summary  
This chapter explains how to compute partial derivatives of multivariable functions by holding all but one variable constant and applying differentiation rules (e.g., power rule and product rule).  

General domain of usage  
Machine learning (e.g., gradient-based optimization)

**A partial derivative** measures how a multivariable function changes with respect to one variable while keeping all other variables constant. It captures the rate of change along a single dimension within a multivariable system.


Definition

## What Are Partial Derivatives?
A **partial derivative** is written using the symbol $$\partial$$ instead of $$d$$ for regular derivatives. If a function $$f(x,y)$$ depends on both $$x$$ and $$y$$, we compute:
 
$$
\frac{\partial f}{\partial x} \lim_{h \rarr 0} \frac{f(x + h, y) - f(x,y)}{h} \\[6pt] 

\frac{\partial f}{\partial y} \lim_{h \rarr 0} \frac{f(x, y + h) - f(x,y)}{h}
$$

When differentiating with respect to one variable, treat all other variables as constants.

Note

## Computing Partial Derivatives

Consider the function:

$$
f(x,y) = x^2y + 3y^2
$$
 
Let's find, $$\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial x$}}$$:

$$
\frac{\partial f}{\partial x} = 2xy
$$

- Differentiate with respect to $$x$$, treating $$y$$ as a **constant**.

Let's **compute**, $$\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial y$}}$$: 

$$
\frac{\partial f}{\partial y} = x^2 + 6y
$$

- Differentiate with respect to $$y$$, treating $$x$$ as a **constant**.

Consider the function:

$$
f(x,y) = 4x^3y + 5y^2
$$

Now, compute the partial derivative with respect to $$y$$.


Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Introductions to Partial Derivatives

What Are Partial Derivatives?

Computing Partial Derivatives