#### 1.1 Partial Differentiation

Here we shall discuss differentiation of a function containg more than one independent variable. To start with, we shall consider the case of a function *u* which contains two independent variables *x* and *y*. The derivative of *u*, when *x* varies and *y* remains constant, is called *the partial derivative of **u* with respect to x. Similarly, *the partial derivative of u with respect to y* is the differential coefficient of *u* with respect to *y*, when *x* remains constant.

*Notation:* Partial derivatives of *u* with respect to *x* and *y* are denoted by ∂*u/*∂*x* and ∂*u*/∂*y*. Similar notations are used for higher derivatives. The formal definition of ∂*u/*∂*x*, when *u* = *f* (*x*, *y*), is

Similarly,

**Example 1:** Find the partial derivative of ...