. Partial Derivatives Partial derivative of a function of several variables is its derivative with respect to one of those variables, keeping other variables as constant. In this section, we will restrict our study to functions of two variables and their derivatives only. Let u f x y ( , ) be a function of two independent variables x and y .
- - The derivative of u with respect to x when x varies and y remains constant is called the partial derivative of u with respect to x , denoted by ∂ ∂ u x (or) u x and is defined as u = lim T , ) ( , ) f x x y f x y T T provided the limit exists. Here ∆ x is a small change in x The derivative of u with respect to y , when y varies and x remains constant is called the partial derivative of u with respect to y , denoted by ∂ ∂ u y (or) u y and is defined as u = lim $ O ( , ( , ) f x y f x y T T provided the limit exists. Here ∆ y is a small change in y . u is also written as x f ( x , y ) (or) x .
Similarly ¶ ¶ u y is also written as ¶ ¶ y f ( x , y ) (or) y The process of finding a partial derivative is called partial differentiation. . . Successive partial derivatives Consider the function u f x y ( , ) .
From this we can find u and u . If u and u are functions of x and y , then they may be differentiated partially again with respect to either of the independent variables, ( x or y ) denoted by u u y x u , y x u These derivatives are called second order partial derivatives. Similarly, we can find the third order partial derivatives, fourth order partial derivatives