Lecture ¹3

# Lecture ¹3. The higher-order partial derivatives and differentials. Taylor’s formula

# Exercises for Lecture #3

Content

1.7 Higher-Order Derivatives and Differentials

1.8 Taylor's Formula for a Function of Several Variables

The purpose of work:

The acquaintance with the higher-order partial derivatives and differentials of functions of many variables

Theoretical information and examples

1.7 Higher-Order Derivatives and Differentials

1.7.1 Higher order partial derivatives. The second partial derivatives of a function z = f(x, y) are the partial derivatives of its first partial derivatives.

For second derivatives, we use the notation

Derivatives of order higher than second are similarly defined and denoted.

If the partial derivatives to be evaluated are continuous, then the result of repeated differentiation is independent of the order in which it is performed.

Example 1. Find the second partial derivatives of the function

Solution. First find the first partial derivatives:

Now differentiate a second time:

We note that the so-called "mixed" partial derivative may be found in different ways, namely:

1.7.2 Higher-order differentials. The second differential of a function z=f(x, y) is the differential of the (first-order) differential of this function:

.

We define similarly the differentials of a function z of order higher than two, for example:

and, generally,

.

If z =f(x, y), where x and y are independent variables, then the second differential of the function z is computed from the formula

In general, the following symbolic formula holds true:

it is formally expanded by the binomial law.

If z = f(x, y), where the arguments x and y are functions of one or several independent variables, then

If x and y are independent variables, then, and (2) becomes identical with (1).

Example 2. Find the total differentials of first and second orders of the function

.

Solution/First method. We have

whence

Moreover, we have

whence

Second method. Differentiating, we find

Differentiating again and remembering that dx and dy do not depend on x and y, we find

1.8 Taylor's Formula for a Function of Several Variables

Let the function f(x, y) have continuous partial derivatives of all orders up to and including the (n+l)-th in the neighbourhood of a point (a, b). Then, Taylor's formula will hold in the neighbourhood under consideration:

Where

In a different notation

or

The particular case of (1), when a = b = 0, is called Maclaurin's formula.

Similar formulae hold for functions of three and larger numbers of variables.

Example. Find the increment of the function

when passing from the values x = l, y = 2 to the values

Solution. The required increment may be found by applying (2). First of all, calculate the successive partial derivatives and their values at the given point (1, 2):

All subsequent derivatives are identically zero. Substituting these results into (2), we obtain:

Control questions

1. What are the second partial derivatives of a function z = f(x, y)?

2. In what case the result of repeated differentiation is independent of the order in which it is performed?

3. What is the second differential of a function z=f(x, y)?