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CHAPTER 5: PARTIAL DERIVATIVES

Functions of n Independent Variables

Suppose D is a set of n-tuples of real numbers . A real valued function f on D is a rule that assigns a unique (single) real number

to each element in D. The set D is the function’s domain. The set of w-values taken on by f is the function’s range. The symbol w is the dependent variable of f, and f is said to be a function of the n independent variables to . We also call the the function’s input variables and call w the function’s output variable.

Level Curve, Graph, surface of Functions of Two Variables

The set of points in the plane where a function has a constant value is called a level curve of f. The set of all points in space, for in the domain of f, is called the graph of f. The graph of f is also called the surface .

Functions of Three Variables

The set of points in space where a function of three independent variables has a constant value is called a level surface of f.

Example: Attend lecture.

Partial Derivatives of a Function of Two Variables

Definition: Partial Derivative with Respect to x

The partial derivative of with respect to x at the point is

provided the limit exists.

Definition: Partial Derivative with Respect to y

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The partial derivative of with respect to y at the point is

provided the limit exists.

Example:

1. Find the values of and at the point if .

2. Find if .

3. Find and if .

Functions of More Than Two Variables

Example:

1. Let . Find , and at .

2. Let . Find , and .

Second-Order Partial Derivatives

When we differentiate a function twice, we produce its second-order derivatives.These derivatives are usually denoted by

“ d squared fdx squared “ or “f sub xx “

“ d squared fdy squared “ or “f sub yy “

“ d squared fdx squared “ or “f sub xx “

“ d squared fdxdy squared “ or “f sub yx “

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“ d squared fdydx squared “ or “f sub xy “

The defining equations are

,

and so on. Notice the order in which the derivatives are taken:

Differentiate first with respect to y, then with respect to x.

Means the same thing.

Example:

1. Let . Find , , and .

2. If , find , , and .

The Chain Rule

Chain Rule for Functions of Two Independent Variables

If has continuous partial derivatives and and if , are differentiable functions of t, then the compose is a differentiable function of t and

or

.

Example:

Use the chain rule to find the derivative of , with respect to t along the path

, . What is the derivative’s value at ?

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Chain Rule for Functions of Three Independent Variables

If is differentiable and x, y and z are differentiable functions of t, then w is a differentiable function of t and

.

Example:

Find if , , , .

Chain Rule for Two Independent Variables and Three Intermediate Variables

Suppose that , , , and . If all four functions are differentiable, then w has partial derivatives with respect to r and s, given by the formulas

Example:

Express and in terms of r and s is

, , , .

If , , and , then

and

Example:

Express and in terms of r and s if

, , .

If and , then

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and .

PROBLEM SET: CHAPTER 5

1. Sketch and name the surfaces

(a) (e)(b) (f)(c) (g)

(d) (h)

2. Find and .

(a)

(b)

(c)(d)(e)(f)

3. Find , and .

(a)

(b)

(c)(d)

4. Find all the second-order partial derivatives of the following functions.

(a)(b)(c)(d)

5. Verify that .

(a) (c)

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(b) (d)

6. In the following questions, (a) express as a function of t, both by using

the Chain Rule and by expressing w in terms of t and differentiating directly with

respect to t. The (b) evaluate at the given value of t.

(i) , , ; .

(ii) , , , .

7. In the following questions, (a) express and as a functions of u and v both

by using the Chain Rule and by expressing z directly in terms of u and v before

differentiating. Then (b) evaluate and at the given point .

(i) , , ;

(ii) , , ;

ANSWERS FOR PROBLEM SET: CHAPTER 5

2. (a)

(b)

(c)

(d)

(e)

(f)

3. (a) , ,

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(b) , ,

(c) , ,

(d) , ,

4. (a) , ,0y

f2

2

(b) ,

(c) ,

(d) ,

5. (a) and

(b) and

x

1

y

1

yx

w2

(c)

and

(d)

and

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6. (i) (a) (b)

(ii) (a) (b) 1

7. (i) (a)

(b)

(ii) (a)

(b)