Partial Derivative Of Functions

In calculus, partial derivatives are a fundamental concept when dealing with functions of multiple variables. They extend the idea of a derivative to multivariable functions and are crucial in various fields such as engineering, physics, economics, and machine learning.

Definition of Partial Derivative

The partial derivative of a function measures how the function changes as one of its variables changes, while keeping all other variables constant. This is particularly useful for analyzing functions with more than one independent variable.

If f(x, y) is a function of x and y, the partial derivatives of f with respect to x and y are denoted as:

∂f/∂x and ∂f/∂y 

Notation for Partial Derivatives

Several notations are commonly used for partial derivatives:

• Leibniz Notation:∂f/∂x, ∂f/∂y
• Compact Notation:f_x, f_y

For higher-order partial derivatives:

• ∂2f/∂x2: Second-order derivative with respect to x
• ∂2f/∂x∂y: Mixed derivative (first with respect to y, then x)

Formulas for Partial Derivatives

To compute a partial derivative:

1. Differentiate the function with respect to one variable while treating all other variables as constants.
2. Apply the standard differentiation rules:
   • Power Rule:∂/∂x [x^n] = nx^{n-1}
   • Constant Rule:∂/∂x [c] = 0 (where c is constant)
   • Sum Rule:∂/∂x [f + g] = ∂f/∂x + ∂g/∂x
   • Product Rule:∂/∂x [uv] = u(∂v/∂x) + v(∂u/∂x)
   • Chain Rule: For f(g(x)), ∂f/∂x = (∂f/∂g)(∂g/∂x)

Examples of Partial Derivatives

Example 1: Simple Function

Given f(x, y) = x^2y + 3xy^2, find ∂f/∂x and ∂f/∂y.

Solution:

• Partial derivative with respect to x:

∂f/∂x = 2xy + 3y^2 

∂f/∂y = x^2 + 6xy 

Example 2: Higher-Order Partial Derivatives

For f(x, y) = x^2 + 3xy + y^3, find the second-order partial derivatives ∂2f/∂x2, ∂2f/∂y2, and ∂2f/∂x∂y.

Solution:

• ∂2f/∂x2 = 2
• ∂2f/∂y2 = 6y
• ∂2f/∂x∂y = 3

Example 3: Application in Real Life

In economics, the utility function U(x, y) = 5x^{0.5}y^{0.5} represents the satisfaction a consumer gets from consuming x units of good 1 and y units of good 2. Find the marginal utility of x (partial derivative with respect to x).

Solution:

∂U/∂x = 2.5x^{-0.5}y^{0.5} 

Applications of Partial Derivatives

Partial derivatives are used in:

• Optimization problems: Finding maximum and minimum values of multivariable functions.
• Physics: Describing wave functions, heat transfer, and fluid dynamics.
• Economics: Computing marginal costs and utility.
• Machine Learning: Backpropagation and gradient descent algorithms.

Conclusion

Partial derivatives provide a framework to analyze multivariable functions by focusing on the rate of change with respect to one variable at a time. Mastering the computation of partial derivatives, along with their applications, is essential for solving complex problems in science, engineering, and beyond.