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.