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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 (-)