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 $$y$$.


This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Introductions to Partial Derivatives

What Are Partial Derivatives?

Computing Partial Derivatives