Math 1431

Section 14839 M TH 4:00 PM-5:30 PM Online

Susan Wheeler

swheeler@math.uh.edu

Office Hours: 5:30 - 6:15 pm M Th Online or by appointment

Wed 6:00 – 7:00 PM Online

Class webpage: http://www.math.uh.edu/~swheeler/math1431.html

The Definition of the Derivative

A function f (x) is differentiable at x if and only if

( ) ( )f f

lim→

+ −h 0

x h x

h

exists. In this case, we denote

( ) ( ) ( )f ff ' lim

→

+ −=h 0

x h xx

h

and we refer to as the derivative of f at x.

( )xf '

If f is differentiable at x = a, then f is continuous at x = a. Not every continuous function is differentiable. A function is not differentiable at 1. points of discontinuity 2. cusps 3. sharp turns (corners)

How can we use the derivative to find the slope of the normal line to the

graph of f (x) at x = a?

The normal line to the graph at x = a is the perpendicular line to the graph at x = a .

That is:

The normal line is perpendicular to the tangent line at x = a.

Algebraic Properties of the Derivative

Differentiation Formulas

Section 2.2

If f and g are differentiable and c is a scalar, then f + g, f – g and (c f ) are differentiable. Furthermore,

Derivative of the sum is the sum of the derivatives.

Derivative of the difference is the difference of the derivatives.

And the derivative of any scalar times a function is the scalar times the derivative of the function.

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( )( ) ( )

d d dx x x x

dx dx dx

d d dx x x x

dx dx dx

d dc x c x

dx dx

f g f g

f g f g

f f

+ = +

− = −

=

=d8

dx

=dx

dx

( ) =d5x

dx

( )+ =d5x 2

dx

Power Rule

( ) −=n n 1dx nx

dx , n ≠ 0

Find the derivative of each.

( )f = 2x x

( )f = 3x x

( )f = −5 2x x x

( )f = + −4 3x 3x 2x 4x

( )f = =12x x x

( )f = +9 57 7x x x

f x( ) = 1

x2

Higher Order Derivatives

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

4

2 3 4

2 3 4

x x x x

d d d dx x x x

dx dx dx dx

f ' , f '' , f ''' , f

f , f , f , f

Determine

Determine

( )2

3 22

d3x 5x 2x 1

dx− + −

( )3

8 53

d3x 2x 3x 5

dx+ − −

Trig Derivatives:

ddxsin x = cos x

ddxcos x = −sin x

ddxtan x = sec2 x

ddxcot x = −csc2 x

ddxsec x = sec x ⋅ tan x

ddxcsc x = −csc x ⋅cot x

MEMORIZE THESE!

Products and Quotients

If f and g are differentiable then f •g and f / g are differentiable.

furthermore

ddx

f x( ) • g x( )( ) = f '(x )g(x ) + f (x )g '(x )

ddx

f x( )g x( )

⎛

⎝⎜

⎞

⎠⎟ =

g x( ) f ' x( ) − f x( )g ' x( )g x( )( )2

Find the derivative of each. ( ) ( )( )f = + + =x 5x 2 x 1

( ) ( )( )f = + − =4x 3x 5 2x x

( ) ( )( )f = − + + =2 3x x 2x 1 x 1

Find the derivative of each.

( )f =+2x

xx 1

( )f −=−

2x 4x

x 3

( )f =+1

xx 1

( )f = 1xx

( )f =2

1x

x

The Chain Rule Find the derivative of a) = 2y 5x b) ( )= + 2y 2x 1 c) ( )= + 15y 2x 1

Recall: Composite functions are functions within functions. They are written ( )( ( )) ( )f g x or f g xo Example: If f (x) = 3x – 4 and g(x) = x2 , then f (g (x))= and g (f (x))= To find the derivative of composite functions, we use the chain rule.

Chain Rule Let f (x) and g(x) be separate functions of x and let y = f (g (x)), then ( )( ) ( )' f ' g g'= ⋅y x x Examples: 1) ( ) ( )h = +

15x 2x 1

2) ( )= −5

y x 1 3) ( )= + +

32y x 6x 1

4) = +2y x 3 5) ( )= +

4y 6 3x 2

6) ⎛ ⎞= +⎜ ⎟⎝ ⎠

32

2

1y x

x

7) ( )f ⎛ ⎞= ⎜ ⎟+⎝ ⎠

4

2

xx

2x 1

If G(x) = f ( v ( x) ), find G’ (1) .

v(1) = 2 f ’(1) = 3 f ’(2) = –6 v ’(1) = 7