PARTIAL DIFFERENTIATION SEBASTIAN VATTAMATTAM
1. Partial Derivatives and Chain Rules Definition 1.1. Let f be a function of several variables. Its derivative with respect to one of those variables, keeping others constant, is called a partial derivative. The partial derivative of function f with respect to x is denoted by ∂f ∂x or fx . Definition 1.2. Let f be a function of x, y. ∂f ∂f ∂x , ∂y are called the first order partial derivatives of f. See Figure 1 The second order partial derivatives are ∂ 2f ∂x2 ∂ 2f ∂y 2 ∂ 2f ∂x∂y ∂ 2f ∂y∂x
∂ ∂f ( ) ∂x ∂x ∂ ∂f = ( ) ∂y ∂y ∂ ∂f = ( ) ∂x ∂y ∂ ∂f = ( ) ∂y ∂x =
Example 1.3. f (x, y) = x2 sin y 1