AP Calculus Derivatives and Antiderivatives

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.

Derivative of Sin

Cos

Derivative of Cos

-Sin

Derivative of Sin(ax)

acos(ax)

Derivative of Cos(ax)

-asin(ax)

Antiderivative of Cos

Sinx + C

Antiderivative of Sin

-Cosx +C

Antiderivative of Sin(ax)

-1/aCos(ax) +C

Antiderivative of Cos(ax)

1/aSin(ax) +C

Derivative of e^x

e^x

Antiderivative of e^x

e^x+c

Derivative of b^x

b^x * ln(b)

Antiderivative of b^x

b^x/ln(b) +c

Derivative of ln(x)

1/x

Antiderivative of 1/x

ln(x) +c

Derivative of log(b)(X)

1/xln(b)

Exponential Growth Equation

y= amount*e^(rate*time)

Finding exponential growth equation

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

First order DE

First derivative equation

Second order DE

Second derivative equation

Power Rule

exp(x)^exp-1

Chain Rule

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

Product Rule

f*Ds + s*Df

Quotient Rule

LodHi-HidLow/Low^2

Derivative of Inverse Function

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

Inverse of Sin

Sin^-1

Derivative of arcsin

1/ √(1-x^2)

Inverse of cos

cos^-1

Derivative of arccos

-1/√(1-x^2)

Inverse of tan

tan^-1

Derivative of arctan

1/1+x^2

Inverse of sec

sec^-1

Derivative of arcsec(x)

1/ IxI √(x^2+1)

Inverse of csc

csc^-1

Derivative of arccsc

-1/ IxI √(x^2+1)

Inverse of cot

cot^-1

Derivative of arccot

-1/ 1+x^2

F's graph has (-3

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

Inverses in 2nd Quadrant

arccot

Inverses in 4th Quadrant

arctan

Domain of arcsin and arccos

(-1

Domain of arctan and arccot

(-inf

Domain of arcsec and arccsc

IxI > or equal to 1

Range of arcsin and arctan

(-pi/2

Range of arccos and arccot

(0

Range of arcsec

(0

Range of arccsc

(-pi/2

d^2y/dx^2

Leibniz Notation for Second Derivative

dy/dx

Leibniz Notation for First Derivative

Derivative of tan

sec^2(x)

Antiderivative of sec^2(x)

tan(x) + C

Derivative of cot

-csc^2(x)

Antiderivative of csc^2(x)

-cot(x) + C

Derivative of sec(x)

sec(x)tan(x)

Antiderivative of sec(x)tan(x)

sec(x) + C

Derivative of csc

-csc(x)cot(x)

Antiderivative of csc(x)cot(x)

-csc(x) + C

AOR is Higher or to Right

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

AOR is Lower or to Left

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

IBP

U and dv are in regular equation

Pumping Fluids work

Area

Sin^2x

1/2 (1-cos(theta))

Cos^2x

1/2 (1+Cos(theta))

a^2-f(x)^2

x=a*sin(theta)

a^2+f(x)^2

x=a*tan(theta)

f(x)^2-a^2

x=a*sec(theta)

sinAcosB

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

sinAsinB

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

cosAcosB

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

Trig ID for Sec and Tan

Sec^2-Tan^2=1

integral of tanx

ln|secx|

integral of sec

ln |sec + tan|

sin(2x)

2sinxcosx

cos(2x)

cos^2x-sin^2x

Average Value of a Function

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

Pumping water from top

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

Pumping Water from bottom

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

Disk and Washer are

Perpendicular to AOR

Shell is

Parallel to AOR

if PF denom has no powers

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

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

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

if PF has squares inside denom

A+Bx/(Term^2)

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

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)

last-ditch effort

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

For PF...

numerator has to be less than denom

Trapezoid Rule

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

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

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

Most accurate ways for approxing integrals

simpson's (if even)

Indeterminate Forms

0/0 and +-infinity/+-infinity

Limits for Lhop

put them on both numerator and denominator

indeterminate product

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

indeterminate differences

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

if denom approaches zero

fraction goes inf

if denom goes to inf

fraction goes to zero

indeterminate powers

inf^0

only way 0^0 is 1

if both functions equal zero exactly

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

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

if graph given for lhop

use slope of tangent line

improper infinite integrals

break it up