Lecture ¹2

# Lecture ¹2. The partial derivatives and differentials of functions of many variables

The necessary and sufficient conditions of function’s differentiability. The partial derivative of composite function. The directional derivative. Gradient.

# Exercises for Lecture 2 Examples

The purpose of work:

The acquaintance with the partial derivatives and differentials of functions of many variables.

Theoretical information and examples

Content

1.3 Partial Derivatives

1.4 Total Differential of a Function

1.5 Differentiation of Composite Functions

1.6 Derivative in a Given Direction and the Gradient of a Function

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## 1.3 Partial Derivatives

Let , and let .

1.3.1 Definition of a partial derivative. If z = f(x, y), then assuming, for example, that y is constant, we get the derivative

which is called the partial derivative of the function z with respect to the variable x. In a similar fashion, we define and denote the partial derivative of the function z with respect to the variable y. It is obvious that, in order to find partial derivatives, one can use the ordinary differentiation formulae.

Example 1. Find the partial derivatives of the function

Solution. Regarding y as a constant, we get

Similarly, keeping x constant, we have

Example 2. Find the partial derivatives of the function of three arguments:

Solution.

1.3.2 Euler's theorem. A function f(x, y) is called a homogeneous function of degree n if for every real factor k

A rational integral function will be homogeneous if all its terms are of one and the same degree.

For a homogeneous differentiable function of degree n, one has Euler's theorem:

## 1.4. Total Differential of a Function

1.4.1 Total Increment of a function. The total increment of a function z=f(x, y) is the difference

1.4.2 The total differential of a function. The total (or exact) differential of a function z = f(x, y) is the principal part of the total increment Dz, which is linear with respect to the increments in the arguments Dx and Dy.

The difference between the total increment and the total differential of a function is an infinitesimal of higher order compared with

A function definitely has a total differential if its partial derivatives are continuous. If a function has a total differential, then it is called differentiable. The differentials of independent variables coincide with their increments, that is

dx = Dx and dy = Dy.

The total differential of the function z = f(x, y) is computed from the formula

Similarly, the total differential of a function of three arguments u =f(x,y,z) is computed from

Example 1. Find for the function

the total increment and total differential.

Solution.

Here, the expression

is the total differential of the function, while

is an infinitesimal of higher order compared with

Example 2. Find the total differential of the function

Solution.

1.4.3 Applying the total differential of a function to approximate calculations. For sufficiently small |Dx| and |Dy|, and hence for sufficiently small

we have for a differentiable function z = f(x, y) the approximate equality Dz » dz or

Example 3. The height of a cone is H=30cm, the radius of its base R=10cm. How will the volume of the cone change if we increase H by 3 mm and diminish R by 1 mm?

Solution. The volume of the cone is

We replace the change in volume approximately by the differential

Example 4. Compute 1.023.01 approximately.

Solution. Consider the function z = xy. The required number may be considered to be the increased value of this function when x = 1, y = 3, Dx = 0.02, Dy = 0.0l. The initial value of the function is z = 1,

Hence,

## 1.5 Differentiation of Composite Functions

1.5.1 Case of one Independent variable. If z = f(x, y) is a differentiable function of the arguments x and y, which, in turn, are differentiable functions of an independent variable t,

then the derivative of the composite function x =f[j(t), y(t)] may be computed from the formula

In particular, if t coincides with one of the arguments, for instance x, then the "total" derivative of the function z with respect to x will be

Example 1. Find dz/dt of

where x = cos t, y = t2.

Solution. By (1), we have

Example 2. Find the partial derivative and the total derivative dz/dx if

where

Solution. By (2), we obtain

1.5.2 The case of several independent variables. If z is a composite function of several independent variables, for instance, z =f(x, y), where x=j(u, v), y=y(u, v) (u and v being independent variables), then the partial derivatives with respect to u and v are given by

In all the cases considered applies the formula

(the invariance property of a total differential).

Example 3. Find and if z = f(x, y), where x = uv, y = u/v.

Solution. Applying the above formulae, we find

Example 4. Show that the function z = j(x2 + y2) satisfies the equation

Solution. The function j depends on x and y via the intermediate argument , hence

Substituting the partial derivatives into the left hand side of the equation, we get

i.e., the function z satisfies the given equation.

## 1.6 Derivative in a Given Direction and the Gradient of a Function

1.6.1 The derivative of a function in a given direction. The derivative of a function z = f(x, y) in the direction is

where f(P) and f(P1) are values of the function at the points P and P1. If the function z is differentiable, then

where a is the angle formed by the vector l with the x-axis (Fig.5).

In a similar fashion, we define the derivative in a given direction l for a function of three variables u = f(x, y, z). In this case,

where a, b, g are the angles between the direction l and the corresponding co-ordinate axes. The directional derivative characterizes the rate of change of the function in the given direction.

Example 1. Find the derivative of the function z = 2x2 - 3y2 at the point P(l, 0) in a direction which makes an angle of 120◦ with the x-axis.

Solution. Find the partial derivatives of the given function and their values at the point P:

Here

The minus sign indicates that the function diminishes at the given point and in the given direction.

1.6.2 The gradient of a function. The gradient of a function z = f(x, y) is a vector the projections of which onto the co-ordinate axes are the corresponding partial derivatives of the given function:

The gradient of a function at each point is directed along the normal to the corresponding level line of the function. The direction of the gradient of the function at a given point is the direction of the maximum rate of increase of the function at this point, i.e., when l = grad z, the derivative takes on its largest value equal to

In a similar fashion, we define the gradient of the function of three variables u=f(x, y, z):

The gradient of a function of three variables is directed at each point along the normal to the level surface passing through this point.

Example 2. Find and construct the gradient of the function z = x2y at the point P(1,1).

Solution. Compute the partial derivatives and their values at the point P:

hence grad z = 2i + j , (Fig. 6).

Control questions:

1. What we call the first order partial derivative of function of several variables?

2. What is the total differential of function?

3. What is the first order differential of function of several variables?

4. What is the second order partial derivative of function of several variables?

5. What is the second order differential of function of several variables?

6. What is the gradient of function of several variables?

7. What is the directional derivative of function of several variables?