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Flashcards for all of the most common derivatives and antiderivatives that will likely appear during the AP Exam. Most of these are important to memorize for ease of use.

1

Derivative of Sin

Cos

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2

Derivative of Cos

-Sin

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3

Derivative of Sin(ax)

acos(ax)

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4

Derivative of Cos(ax)

-asin(ax)

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5

Antiderivative of Cos

Sinx + C

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6

Antiderivative of Sin

-Cosx +C

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7

Antiderivative of Sin(ax)

-1/aCos(ax) +C

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8

Antiderivative of Cos(ax)

1/aSin(ax) +C

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9

Derivative of e^x

e^x

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10

Antiderivative of e^x

e^x+c

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11

Derivative of b^x

b^x * ln(b)

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12

Antiderivative of b^x

b^x/ln(b) +c

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13

Derivative of ln(x)

1/x

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14

Antiderivative of 1/x

ln(x) +c

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15

Derivative of log(b)(X)

1/xln(b)

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16

Exponential Growth Equation

y= amount*e^(rate*time)

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17

Finding exponential growth equation

y'(0) = rate*y and y(0) = amount

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18

First order DE

First derivative equation

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19

Second order DE

Second derivative equation

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20

Power Rule

exp(x)^exp-1

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21

Chain Rule

f '(g(x)) * g'(x)

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22

Product Rule

f*Ds + s*Df

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23

Quotient Rule

LodHi-HidLow/Low^2

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24

Derivative of Inverse Function

g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x) (SAME VICE-VERSA)

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25

Inverse of Sin

Sin^-1

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26

Derivative of arcsin

1/ √(1-x^2)

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27

Inverse of cos

cos^-1

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28

Derivative of arccos

-1/√(1-x^2)

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29

Inverse of tan

tan^-1

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30

Derivative of arctan

1/1+x^2

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31

Inverse of sec

sec^-1

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32

Derivative of arcsec(x)

1/ IxI √(x^2+1)

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33

Inverse of csc

csc^-1

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34

Derivative of arccsc

-1/ IxI √(x^2+1)

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35

Inverse of cot

cot^-1

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36

Derivative of arccot

-1/ 1+x^2

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37

F's graph has (-3

what does the inverse of F (F^-1) have?

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38

Inverses in 2nd Quadrant

arccot

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39

Inverses in 4th Quadrant

arctan

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40

Domain of arcsin and arccos

(-1

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41

Domain of arctan and arccot

(-inf

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42

Domain of arcsec and arccsc

IxI > or equal to 1

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43

Range of arcsin and arctan

(-pi/2

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44

Range of arccos and arccot

(0

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45

Range of arcsec

(0

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46

Range of arccsc

(-pi/2

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47

d^2y/dx^2

Leibniz Notation for Second Derivative

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48

dy/dx

Leibniz Notation for First Derivative

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49

Derivative of tan

sec^2(x)

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50

Antiderivative of sec^2(x)

tan(x) + C

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51

Derivative of cot

-csc^2(x)

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52

Antiderivative of csc^2(x)

-cot(x) + C

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53

Derivative of sec(x)

sec(x)tan(x)

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54

Antiderivative of sec(x)tan(x)

sec(x) + C

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55

Derivative of csc

-csc(x)cot(x)

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56

Antiderivative of csc(x)cot(x)

-csc(x) + C

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57

AOR is Higher or to Right

#-x is in radius for disk/washer or distance for shells

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58# + x is in radius for disk/washer or distance for shells

AOR is Lower or to Left

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59

IBP

U and dv are in regular equation

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60

Pumping Fluids work

Area

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61

Sin^2x

1/2 (1-cos(theta))

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62

Cos^2x

1/2 (1+Cos(theta))

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63

a^2-f(x)^2

x=a*sin(theta)

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64

a^2+f(x)^2

x=a*tan(theta)

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65

f(x)^2-a^2

x=a*sec(theta)

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66

sinAcosB

1/2[sin(A+B)+sin(A-B)]

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67

sinAsinB

1/2[cos(A-B)-cos(A+B)]

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68

cosAcosB

1/2[cos(A+B)+cos(A-B)]

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69

Trig ID for Sec and Tan

Sec^2-Tan^2=1

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70

integral of tanx

ln|secx|

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71

integral of sec

ln |sec + tan|

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72

sin(2x)

2sinxcosx

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73

cos(2x)

cos^2x-sin^2x

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74

Average Value of a Function

1/b-a (integral from a to b) f(x)dx

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75

Pumping water from top

(y + spout distance) Bounds are from top of tank # - Distance tank is filled.

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76

Pumping Water from bottom

(total distance (inc. spout)) - y). Bounds are from zero to amount tank is filled.

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77

Disk and Washer are

Perpendicular to AOR

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78

Shell is

Parallel to AOR

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79

if PF denom has no powers

A/(term) + B/(other term)

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80

if PF denom has a power outside (ex:3)

A/(term) + B/(Term)^2 + C/(Term)^3

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81

if PF has squares inside denom

A+Bx/(Term^2)

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82

if PF has power inside and outside of denom (Ex: (x^2+4)^3)

Ax+B/(x^2+4) + Cx+D/(x^2+4)^2 + Ex+F/(x^2+4)^3

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83

if PF has multiple terms

try and factor the ones that are squared Ex: x^2-5 can be (x+sqrt 5) (x- sqrt 5)

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84

last-ditch effort

make u entire bottom of integral and manipulare ir from there.

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85

For PF...

numerator has to be less than denom

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86

Trapezoid Rule

1/2 ( b-a / n ) (y + 2y + ... + 2y + y)

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87

Simpson's Rule (can only be used when interval # is even)

h/3(1y+4y+2y...2y+4y+1y

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88

Most accurate ways for approxing integrals

simpson's (if even)

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89

Indeterminate Forms

0/0 and +-infinity/+-infinity

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90

Limits for Lhop

put them on both numerator and denominator

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91

indeterminate product

0* +- inf. Flip one and make it Lhop

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92

indeterminate differences

only inf-inf. make them fractions and have common denominator. should be IDF then Lhop.

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93

if denom approaches zero

fraction goes inf

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94

if denom goes to inf

fraction goes to zero

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95

indeterminate powers

inf^0

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96

only way 0^0 is 1

if both functions equal zero exactly

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97

how to solve indeterminate powers problems

y= f(x)^g(x) ans make it log form lny=g(x)*lnf(x). take limit of both sides

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98

if lhop looks complicated after 1-2 times or fractions

Try to simplify it to make it 1 fraction with a num and a denom

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99

if graph given for lhop

use slope of tangent line

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100

improper infinite integrals

break it up

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