Partial Derivatives

Nerd Cafe

What Are Partial Derivatives?

A partial derivative is the derivative of a multivariable function with respect to one variable, while keeping the other variables constant.

Example:

Let

$$f(x,y)=x^{2}y+3xy^{2}$$

Then:

• Partial derivative with respect to x: treat y as constant

• Partial derivative with respect to y: treat x as constant

Step-by-Step: How to Calculate

Let’s calculate the partial derivatives of

$$f(x,y)=x^{2}y+3xy^{2}$$

Step 1: Partial Derivative with Respect to x

• y is constant

• Differentiate:

$$\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}\left( x^{2}y+3xy^{2} \right)=2xy+3y^{2}$$

Step 2: Partial Derivative with Respect to y

• x is constant

• Differentiate:

$$\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left( x^{2}y+3xy^{2} \right)=x^{2}+6xy$$

Python Code with SymPy

import sympy as sp

# Define symbols
x, y = sp.symbols('x y')

# Define the function
f = x**2 * y + 3 * x * y**2

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

# Show results
print("f(x, y) =", f)
print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)

Output

f(x, y) = x**2*y + 3*x*y**2
∂f/∂x = 2*x*y + 3*y**2
∂f/∂y = x**2 + 6*x*y

Geometric Meaning

• For a function f(x,y), the partial derivative ∂f/∂x represents the slope of the surface in the x-direction, holding y constant.

• Likewise, ∂f/∂y is the slope in the y-direction.

Visualize it as slicing a 3D surface either along x or along y, and measuring the slope.

Visualizing Partial Derivatives in Python

import numpy as np
import matplotlib.pyplot as plt

# Create grid
x_vals = np.linspace(-3, 3, 50)
y_vals = np.linspace(-3, 3, 50)
X, Y = np.meshgrid(x_vals, y_vals)

# Function and derivatives
Z = X**2 * Y + 3 * X * Y**2
df_dx = 2*X*Y + 3*Y**2
df_dy = X**2 + 6*X*Y

# Plot f(x, y)
fig = plt.figure(figsize=(12, 4))
ax = fig.add_subplot(131, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_title('f(x, y)')

# ∂f/∂x
ax = fig.add_subplot(132, projection='3d')
ax.plot_surface(X, Y, df_dx, cmap='plasma')
ax.set_title('∂f/∂x')

# ∂f/∂y
ax = fig.add_subplot(133, projection='3d')
ax.plot_surface(X, Y, df_dy, cmap='coolwarm')
ax.set_title('∂f/∂y')

plt.tight_layout()
plt.show()

Output

Practice Problem

$$f(x,y,z)=x^{2}y+y^{2}z+z^{2}x$$

Find:

$$\frac{\partial f}{\partial x} \; , \; \frac{\partial f}{\partial y} \; , \; \frac{\partial f}{\partial z}$$

Solution (with Python)

import numpy as np
import matplotlib.pyplot as plt

# 3-variable function
x, y, z = sp.symbols('x y z')
f = x**2*y + y**2*z + z**2*x

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

print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)
print("∂f/∂z =", df_dz)

Output

∂f/∂x = 2*x*y + z**2
∂f/∂y = x**2 + 2*y*z
∂f/∂z = 2*x*z + y**2

Keywords

partial derivatives, multivariable calculus, symbolic differentiation, sympy, python math, gradient, calculus, partial derivative examples, multivariable functions, partial derivative python, ∂f/∂x, ∂f/∂y, visualization, surface plot, 3D calculus, partial derivatives visualization, tangent plane, mixed partials, first-order derivatives, mathematical modeling, nerd cafe