# Partial differentiation

## Partial Differentiation Basics

**Partial differentiation**is a concept from multivariate calculus used when dealing with functions of more than one variable.- It involves differentiating a function w.r.t. one variable while keeping all other variables constant. It calculates the change in the function when one variable changes and all others are held constant.
- For a function f(x, y), the partial derivatives with respect to x and y are denoted by
**∂f/∂x**and**∂f/∂y**respectively. - The notation
**∂**(representing the Greek letter ‘delta’) is used to demonstrate that the differentiation is partial, as opposed to total.

## Calculation of Partial Derivatives

- Any constants or terms not involving the variable of interest are treated as constants during the differentiation.
- The standard rules of differentiation (power rule, product rule, quotient rule, chain rule) apply for
**partial differentiation**, with additional care for the variable being treated as a constant. - For example, if f(x, y) = xy^2 + 3y + 2, the partial derivative
**∂f/∂x**= y^2 (since y is treated as constant while differentiating with respect to x).

## First and Higher-Order Partial Derivatives

- The derivative of a first order partial derivative is called a
**second order partial derivative**. Higher order derivatives can be calculated similarly. - Notation for second order derivatives: ∂²f/∂x², ∂²f/∂y² for second derivatives w.r.t x or y. ∂²f/∂x∂y or ∂²f/∂y∂x denotes mixed partial derivatives.
- The
**order of differentiation matters**for mixed partial derivatives, and only equals if the function is said to be**‘Clairaut’s theorem’**satisfied.

## Interpretations of Partial Derivatives

- A
**partial derivative**gives the rate at which the function is changing with respect to the variable of interest, while keeping all other variables constant. - This is useful in many fields, such as physics or economics, where one might want to understand how a system behaves when one parameter changes but all others stay the same.
- The
**gradient vector**, which combines all first-order partial derivatives of a function, points in the direction of the greatest rate of increase of the function.

## Applications of Partial Differentiation

- Partial derivatives are used to find
**local minimum**and**maximum**of functions of multiple variables - very useful in optimization problems. - The method of
**Lagrange multipliers**, which requires partial derivatives, allows us to find the local maximum or minimum of a function subject to constraints. - In physics, partial differentiation plays a key role in the formulation and solutions of differential equations that describe physical phenomena.