Summary  
This chapter shows how to use Python’s SymPy library to define symbolic multivariable functions, compute their partial derivatives with respect to each variable, evaluate those derivatives at given points, and printing the results.

General domain of usage  
Optimization

In this video, you will learn how to compute **partial derivatives** of multivariable functions using Python. They are essential in **optimization, machine learning, and data science** for analyzing how a function changes with respect to one variable while keeping others constant.


## 1. Defining a Multivariable Function
```
x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2
```
- Here, we define $$x$$ and $$y$$ as symbolic variables;
- We then define the function $$f(x, y) = 4x^3y + 5y^2$$.


## 2. Computing Partial Derivatives
```
df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)  
```
- `sp.diff(f, x)` computes $$\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial x$}}$$ while treating $$y$$ as a constant;
- `sp.diff(f, y)` computes $$\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial y$}}$$ while treating $$x$$ as a constant.


## 3. Evaluating Partial Derivatives at (x=1, y=2)
```
df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})
```
- The `.subs({x: 1, y: 2})` function substitutes $$x=1$$ and $$y=2$4 into the computed derivatives;
- This allows us to **numerically evaluate** the derivatives at a specific point.


## 4. Printing the Results

We print the **original function**, its **partial derivatives**, and their **evaluations** at $$(1,2)$$.

import sympy as sp

x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2

df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)

df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})

print("Function: f(x, y) =", f)
print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)
print("∂f/∂x at (1,2) =", df_dx_val)
print("∂f/∂y at (1,2) =", df_dy_val)

What will `sp.diff(f, y)` return for given function?

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.

Implementing Partial Derivatives in Python

1. Defining a Multivariable Function

2. Computing Partial Derivatives

3. Evaluating Partial Derivatives at (x=1, y=2)

4. Printing the Results