The Calculus AB BibleThe 2nd most important book in the world ™

Derivative FormulasDerivative Notation:

For a function f(x), the derivative would be

Leibniz's Notation:

For the derivative of y in terms of x, we write

For the second derivative using Leibniz's Notation:

Product Rule:

Quotient Rule:

Chain Rule:

y f x

dy

dxn f x f x

n

n

( ( ))

( ( )) ( ' ( ))1

y x

dy

dxx x

dy

dxx x

( )

( )

( )

2 3

2 2

2 2

1

3 1 2

6 1

Natural Log

y f x

dy

dx f xf x

ln( ( ))

( )' ( )

1

y x

dy

dx xx

dy

dx

x

x

ln( )2

2

2

1

1

12

2

1

Power Rule:

y x

dy

dxax

a

a

1

y x

dy

dxx

2

10

5

4

Constant with a Variable Power:

y a

dy

dxa a f x

f x

f x

( )

( ) ln ' ( )

y

dy

dx

x

x

2

2 1ln

y

dy

dxx

x

x

3

3 3 2

2

2

ln

Variable with a Variable power

y f x g x ( ) ( ) Take ln of both sides!

y x

y x

y x x

y

dy

dxx x

xx

dy

dxx x x

xx

x

x

x

sin

sin

sin

ln ln

ln sin ln

cos ln sin

[cos ln sin ]

1 1

1

Implicit Differentiation:Is done when the equation has mixed variables:

x x y y

derivative x xy ydy

dxx y

dy

dx

y xdy

dxy

dy

dxx xy

dy

dx

x xy

y x y

2 2 3 4

3 2 2 3

2 2 3 3

3

2 2 3

5

2 2 3 4 0

3 4 2 2

2 2

3 4

[ ]

Trigonometric Functions:

d

dxx xsin cos

d

dxx xcos sin

d

dxx xtan sec 2

d

dxx x xsec sec tan

Inverse Trigonometric Functions:

y x

dy

dx x

arcsin

1

11

2

y x

dy

dx x

arctan

1

11

2

y x

dy

dx xx

arcsin 4

8

31

14

y x

dy

dx xx

arctan

1

13

62

Integral FormulasBasic Integral

5

5

dx

x C

,where C is an arbitrary constant

x C

Variable with a Constant Power

x dx

x

aC

a

a

1

1

x dx

xC

3

4

4

Constant with a Variable Power

a dx

a

aC

x

x

ln

5

5

5

x

x

dx

C

ln

3

3

2 3

2

2

x

x

dx

C

ln

Fractionsif the top is the derivative of the bottom

1

3

4

4

3

xdx

x dx

x

-unless-

1

xdx

x C

ln| |

x

xdx

x C

3

4

4

11

41

ln| |

Substitution

When integrating a product in which the terms are somehow related, we usually let u= the part in the parenthesis, the part under the radical, the denominator, the exponent, or the angle of the trigonometric function

x x dx u x

du x dx

x x dx

u du

u C

x C

2 2

2 1 2

1 2

3 2

2 3 2

1 1

2

1

22 1

1

21

2

2

31

31

;

( )

( )

/

/

/

/

cos ;

cos

cos

sin

sin

2 2

2

1

22 2

1

21

21

22

x dx u x

du dx

x dx

u du

u C

x C

Integration by Parts

When taking an integral of a product, substitute for u the term whose derivative would eventually reach 0 and the other term for dv.

The general form: uv vdu (pronounced "of dove")

Example:

x e dx

u x dv e dx

du dx v e

x e e dx

xe e C

x

x

x

x x

x x

1

Example 2:

Cxxxxx

xxxxx

xvdxdu

dxxdvxu

xxxx

xvdxxdu

dxxdvxu

xx

sin2cos2sin

cos2cos2[sin

cos2

sin2

sin2sin

sin2

cos

cos

2

2

2

2

2

Inverse Trig Functions

Formulas:

Ca

xxa

arcsin

22

1

Ca

x

a

xa

arctan

1

122

Examples:

Cx

xvax

3arcsin

;3;29

1

Cx

xvax

4arctan

4

1

;4;16

12

More examples:

Cx

x

xvax

2

3arcsin

3

194

3

3

1

3;2;294

1

2

Cx

Cx

x

xvax

4

3arctan

12

14

3arctan

4

1

3

1169

3

3

1

3;4169

1

2

2

Trig Functions

CxdxxCxdxx

CxxxCx

dxx

CxdxxCxdxx

sinlncotcoslntan

sectansec2

2sin2cos

tanseccossin 2

Properties of LogarithmsForm

logarithmic form <=> exponential form

y = loga x <=> ay = x

Log properties

y = log x3 => y = 3 log x

log x + log y = log xy

log x – log y = log (x/y)

Change of Base Law

This is a useful formula to know.

a

xor

a

xxy a ln

ln

log

loglog

Properties of Derivatives

1st Derivative shows: maximum and minimum values, increasing and decreasing intervals, slope of the tangent line to the curve, and velocity

2nd Derivative shows: inflection points, concavity, and acceleration

- Example on the next page -

Example:23632 23 xxxy Find everything about this function

2,3

)2)(3(60

)6(60

3666

2

2

x

xx

xx

xxdx

dy

1st derivative finds max, min, increasing, decreasing

max min(-2, 46) (3, -79)increasing decreasing( , -2] [3, ) (-2, 3)

2

1

)12(60

6122

2

x

x

xdx

yd

2nd derivative finds concavity and inflection points

inflection pt( ½ , -16 ½)concave up concave down( , ½) (1/2, )

Miscellaneous

Newton 痴 Method

Newton 痴 Method is used to approximate a zero of a function

)('

)(

cf

cfc where c is the 1st approximation

Example:If Newton 痴 Method is used to approximate the real root of x3 + x – 1 = 0, then a first approximation of x1 = 1 would lead to athird approximation of x3:

13)('

1)(2

3

xxf

xxxf

3

2

686.86

59

)4

3('

)4

3(

4

3

750.4

3

)1('

)1(1

xorf

f

xorf

f

Separating Variables

Used when you are given the derivative and you need to take the integral. We separate variables when the derivative is a mixture of variables

Example:

If 49ydx

dy and if y = 1 when x = 0, what is the value of y when

x = 3

1?

Cxy

dxy

dy

dxy

dyy

dx

dy

93

9

99

3

4

44

Continuity/Differentiable Problems

f(x) is continuous if and only if both halves of the function have the same answer at the breaking point.f(x) is differentiable if and only if the derivative of both halves of the function have the same answer at the breaking pointExample:

3,96

)(

3,2

xx

xf

xx

66

)('

)3(62

xf

inplugx

- At 3, both halves = 9, therefore, f(x) is continuous

- At 3, both halves of the derivative = 6, therefore, f(x) is differentiable

Useful Information

- We designate position as x(t) or s(t)- The derivative of position x’(t) is v(t), or velocity- The derivative of velocity, v’(t), equals acceleration, a(t).- We often talk about position, velocity, and acceleration when we 池 e

discussing particles moving along the x-axis.- A particle is at rest when v(t) = 0.- A particle is moving to the right when v(t) > 0 and to the left when v(t) < 0

- To find the average velocity of a particle:

b

a

dttvab

)(1

Average Value

Use this formula when asked to find the average of something

b

a

dxxfab

)(1

Mean Value Theorem

NOT the same average value.

According to the Mean value Theorem, there is a number, c, between a and b, such that the slope of the tangent line at c is the same as the slope between the points (a, f(a)) and (b, f(b)).

ab

afbfcf

)()(

)('

Growth Formulas

Double Life Formula: dtyy /0 )2(

Half Life Formula: htyy /0 )2/1(

Growth Formula: kteyy 0

y = ending amount y0 = initial amount t = timek = growth constant d = double life time h = half life time

Useful Trig. Stuff

Double Angle Formulas: xxx cossin22sin

xxx 22 sincos2cos Identities:

xx

xx

xx

22

22

22

csccot1

sectan1

1cossin

cotsin

costan

cos

sin

cscsin

1sec

cos

1

Integration PropertiesArea

b

a

dxxgxf )]()([ f(x) is the equation on top

Volume

f(x) always denotes the equation on top

About the x-axis: About the y-axis:

b

a

b

a

dxxgxf

dxxf

]))(())([(

)]([

22

2

b

a

b

a

dxxgxfx

dxxfx

)]()([2

)]([2

about line y = -1 about the line x = -1

b

a

dxxf 2]1)([ b

a

dxxfx )]()[1(2

Examples:

]2,0[)( 2xxf x-axis: y-axis:

2

0

42

0

22 )( dxxdxx 2

0

2

0

32 2][2 dxxdxxx

about y = -1In this formula f ( x ) or y is the radius of the shaded region. When we rotate about the line y = -1, we have to increase the radius by 1. That is why we add 1 to the radius

2

0

242

0

22 )12(]1[ dxxxdxx

about x = -1In this formula, x is the radius of the shaded region. When we rotate about the line x = -1, we have the increased radius by 1.

2

0

232

0

2 )(2])[1(2 dxxxdxxx

Trapeziodal Rule

Used to approximate area under a curve using trapezoids.

Area )()(2....)(2)(2)(2 1210 nn xfxfxfxfxf

n

ab

where n is the number of subdivisions

Example:f(x) = x2+1. Approximate the area under the curve from [0,2] using trapezoidal rule with 4 subdivisions

750.416

76)4/76(

4

1

5)4/13(2)2(2)4/5(214

1

)2()5.1(2)1(2)5(.2)0(8

02

4

2

0

fffffA

n

b

a

Riemann Sums

Used to approximate area under the curve using rectangles.

a) Inscribed rectangles: all of the rectangles are below the curve

Example: f(x) = x2 + 1 from [0,2] using 4 subdivisions(Find the area of each rectangle and add together)

I=.5(1) II=.5(5/4) III=.5(2) IV=.5(13/4)Total Area = 3.750

b) Circumscribed Rectangles: all rectangles reach above the curve

Example: f(x) = x2 + 1 from [0,2] using 4 subdivisions

I=.5(5/4) II=.5(2) III=.5(13/4) IV=.5(5)Total Area = 5.750

Reading a Graph

When Given the Graph of f’(x)

Make a number line because you are more familiar with number line.

This is the graph of f’(x). Make a number line. - Where f’(x) = 0 (x-int) is where

there are possible max and mins.- Signs are based on if the graph is

above or below the x-axis (determines increasing and decreasing)

min maxx = -4, 3 x = 0

increasing decreasing(-4,0) (3,6] [-6,-4] (0,3)

To read the f’(x) and figure out inflection points and concavity, you read f’(x) the same way you look at f(x) (the original equation) to figure out max, min, increasing and decreasing.

For the graph on the previous page:

inflection ptx = -2, 2, 4, 5

concave up concave down(-6,-2) (2,4) (5,6) (-2,2) (4,5)

Original Source:http://www.geocities.com/Area51/Stargate/5847/index.html

Signs are determined by if f’(x) is increasing (+) and decreasing (-)