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?

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.

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