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 $(1,2)$.