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Calculus Cheat Sheet Derivatives
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Libro Fisica I-Roberto
Portillo
deosx
Calculus Cheat Sheet
Derivatives
Definition and Notation
f ( x + h) - f ( x)
If y = f ( x ) then the derivative is defined to be f ¢ ( x ) = lim
.
h ®0
h
If y = f ( x ) then all of the following are
Di erentiation (Calculus)
Mathematics Question
Bank
= f ( x ) all of the following are equivalent
notations for derivative evaluated at = a .
If
equivalent notations for the derivative.
df
dy
d
f ¢ ( x ) = y¢ =
=
= ( f ( x ) ) = Df ( x )
dx dx dx
f ¢ ( a ) = y¢
x =a
=
df
=
dx
x =a
dy
dx
Mohmmad Khaja Shareef
= Df ( a )
x =a
Interpretation of the Derivative
2. f ¢ ( a ) is the instantaneous rate of
If y = f ( x ) then,
1. m = f ¢ ( a ) is the slope of the tangent
change of f ( x ) at x = a .
line to y = f ( x ) at
3. If f ( x ) is the position of an object at
= a and the
equation of the tangent line at x = a is
time x then f ¢ ( a ) is the velocity of
This website stores data such as cookies to enable essential site
functionality,
as
well
as
marketing,
personalization,
and analytics. Privacy Policy
given by = f ( a ) + f ¢ ( a )( x - a ) .
the object at x = a .
Maxima And Minima
Derivative
Mathematical Analysis
Calculus
Basic Properties and Formulas
( x ) are differentiable functions (the derivative exists), c and n are any real numbers,
1.
( c f )¢ = c f ¢ ( x )
5.
2.
( f ± g )¢ = f ¢ ( x ) ± g ¢ ( x )
6.
3.
( f g )¢ = f ¢ g +
d
dx
d
kuliah1_2015.ppt
x ) = n x - – Power Rule
(
dx
f g ¢ – Product Rule
n
n 1
wira
d
( f ( g ( x ) ) ) = f ¢ ( g ( x ) ) g¢ ( x )
dx
This is the Chain Rule
7.
æ f ö¢ f ¢ g - f g ¢
Related 4.
titles
– Quotient Rule
çg÷ =
2
g
è ø
(c) = 0

If f ( x ) and
Common Derivatives
d
dx
d
dx
d
dx
d
dx
d
dx
d
( x) = 1
dx
d
( sin x ) = cos x
dx
d
a x ) = a x ln ( a )
(
dx
( cot ) = - csc2 x
d
d
( tan ) = sec2 x
1
-1
1- x
1
( sec x ) = sec x tan x
d
( tan
dx
-1
x) =
d
( ln ( x ) ) =
dx
d
2
( cos x ) = -
dx
x
x
e )=e
(
dx
1
sin - x ) =
(
dx
( cos x ) = - sin x
d
( csc x ) = - csc x cot x
1
x
, x>0
1
( ln x ) = x ,
dx
1 - x2
d
1
( log a ( x ) ) =
dx
1 + x2
Major Revision Facts in
Mathematics
B. N. Kumar
x¹0
1
x ln a
, x>0
Partial Derivatives
© 2005 Paul Dawkins
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.
Aien Nurul Ain
Calculus Cheat Sheet
Chain Rule Variants
The chain rule applied to some specific functions.
n
n -1
d
d
1.
éë f ( x ) ùû = n éë f ( x ) ùû f ¢ ( x )
5.
cos éë f ( x )ùû = - f ¢ ( x ) sin éë f ( x ) ùû
dx
dx
d
d
( x)
2.
e
= f ¢ ( x ) e f ( x)
6.
tan éë f ( x ) ùû = f ¢ ( x ) sec 2 éë f ( x ) ùû
dx
dx
d
f ¢( x)
d
7.
( sec [ f ( x)]) = f ¢( x) sec [ f ( x)] tan [ f ( x)]
3.
ln éë f ( x ) ùû =
dx
dx
f ( x)
f ¢( x)
d
-1
d
8.
tan
é
f
x
ù
=
(
)
¢
2
4.
sin éë f ( x ) ùû = f ( x ) cos éë f ( x ) ùû
ë
û
dx
1 + ëé f ( x )ùû
dx
)
(
(
)
(
(
)
(
)
)
(
(
)
d2 f
dx
2
and is defined as
f ¢¢ ( x ) = ( f ¢ ( x ) )¢ , i.e. the derivative of the
first derivative, f ¢ ( x ) .
f
( n)
f
( n)
Andrew Igla
)
Higher Order Derivatives
th
The Second Derivative is denoted as
The n Derivative is denoted as
( 2)
f ¢¢ ( x ) = f ( x ) =
Mathematics 1St First
Order Linear Di erential
Equations 2Nd Second…
( x) =
dn f
dx
n
AP Calculus Cheat Sheet
3d5w2d
and is defined as
¢
( x ) = ( f ( n-1) ( x ) )
, i.e. the derivative of
st
(n-1) derivative, f ( ) ( x ) .
You're Readingthe
a Preview
n -1
Implicit Differentiation
¢ if e
Find
2 x -9 y
+
3
y
2
Upload your documents to download.
= sin ( y ) + 11x . Remember y = y ( x ) here, so products/quotients of x and y
will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to
OR
differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule).
After differentiating solve for ¢ .
Become a Scribd member for full access. Your
¢ + 11 are free.
cos (30
y ) ydays
e 2 x -9 y ( 2 - 9 y¢ ) + 3x 2 y 2 + 2 x3 y y¢ =first
2e
2 x -9 y
(2x
3
- 9 y¢e
2 x -9 y
+ 3x y + 2 x y y¢ = cos ( y ) y¢ + 11
2
2
Þ
3
y¢ =
2 x -9 y
2
y - 9e2 x -9 y - cos ( y ) ) y¢ = 11 - 2Continue
e
- 3x2 yfor
Free
11 - 2e 2 x -9 y - 3x 2 y 2
2
3
y - 9e2 x -9 y - cos ( y )
A-level Maths Revision
(Cheeky Revision
Shortcuts)
Scool Revision
Increasing/Decreasing – Concave Up/Concave Down
Critical Points
x = c is a critical point of f ( x ) provided either
Concave Up/Concave Down
1. If f ¢¢ ( x ) > 0 for all x in an interval I then
1. f ¢ ( c ) = 0 or 2. f ¢ ( c ) doesn’t exist.
fulltext.pdf
మ
f ( x ) is concave up on the interval I.
చం ద ఖ
2. If f ¢¢ ( x ) < 0 for all x in an interval I then
Increasing/Decreasing
1. If f ¢ ( x ) > 0 for all x in an interval I then
f ( x ) is concave down on the interval I.
f ( x ) is increasing on the interval I.
2. If f ¢ ( x ) < 0 for all x in an interval I then
Inflection Points
x = c is a inflection point of f ( x ) if the
f ( x ) is decreasing on the interval I.
= c.
concavity changes at
3. If f ¢ ( x ) = 0 for all x in an interval I then
f ( x ) is constant on the interval I.
© 2005 Paul Dawkins
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.
Calculus Equations And
Answers (Speedy Study
Guides)
Speedy Publishing
You're reading a preview 
Unlock full access (page 3) by
uploading documents or with a
30 Day Free Trial
Cebu - FB 14 Math
Kei Dee
Continue for Free
Calculus Cheat Sheet
Related Rates
Sketch picture and identify known/unknown quantities. Write down equation relating quantities
and differentiate with respect to t using implicit differentiation (i.e. add on a derivative every time
you differentiate a function of t). Plug in known quantities and solve for the unknown quantity.
Ex. A 15 foot ladder is resting against a wall.
Ex. Two people are 50 ft apart when one
The bottom is initially 10 ft away and is being
starts walking north. The angleq changes at
1
0.01 rad/min. At what rate is the distance
pushed towards the wall at 4 ft/sec. How fast
between them changing when q = 0.5 rad?
is the top moving after 12 sec?
We have q ¢ = 0.01 rad/min. and want to find
x¢ . We can use various trig fcns but easiest is,
x¢
sec q =
Þ secq tan q q ¢ =
50
50
We know q = 0.05 so plug in q ¢ and solve.
x¢
sec ( 0.5 ) tan ( 0.5 )( 0.01) =
50
x¢ = 0.3112 ft/sec
Remember to have calculator in radians!
x¢ is negative because x is decreasing. Using
Pythagorean Theorem and differentiating,
x 2 + y 2 = 152 Þ 2 x x¢ + 2 y y¢ = 0
After 12 sec we have x = 10 - 12 ( 14 ) = 7 and
so y = 152 - 72 = 176 . Plug in and solve
for ¢ .
Real Analysis and
Probability: Solutions to
Problems
Robert P. Ash
CNS-GEAS 5 []
Achilles Aldave
You're Reading a Preview
7 ( - 14 ) + 176 y¢ = 0 Þ y¢ =
7
ft/sec
Upload
4 176your documents to download.
Optimization
Sketch picture if needed, write down equation toOR
be optimized and constraint. Solve constraint for
one of the two variables and plug into first equation. Find critical points of equation in range of
variables and verify that
they areamin/max
needed. for full access. Your
Become
Scribd as
member
Ex. We’re enclosing a rectangular field with
Ex. Determine point(s) on y = x 2 + 1 that are
free.
500 ft of fence material and one side first
of the30 days are
closest to (0,2).
field is a building. Determine dimensions that
will maximize the enclosed area.
Shortcuts to College
Calculus Refreshment Kit
Juan Acevedo
Continue for Free
2
2
Minimize f = d 2 = ( x - 0 ) + ( y - 2 ) and the
Maximize A = xy subject to constraint of
x + 2 y = 500 . Solve constraint for x and plug
into area.
A = y ( 500 - 2 y )
x = 500 - 2 y Þ
= 500 - 2 y 2
Differentiate and find critical point(s).
A¢ = 500 - 4 y Þ y = 125
nd
By 2 deriv. test this is a rel. max. and so is
the answer we’re after. Finally, find x.
x = 500 - 2 (125) = 250
constraint is y = x 2 + 1 . Solve constraint for
x 2 and plug into the function.
x2 = y - 1 Þ f = x2 + ( y - 2)
exam
Maria Deborah Vivas Baluran
2
2
= y -1 + ( y - 2) = y2 - 3 y + 3
Differentiate and find critical point(s).
f ¢ = 2y - 3
Þ y = 32
nd
By the 2 derivative test this is a rel. min. and
so all we need to do is find x value(s).
x 2 = 32 - 1 = 12 Þ x = ± 12
The dimensions are then 250 x 125.
The 2 points are then
(
1
2
)
(
, 32 and -
1
2
, 32
)
© 2005 Paul Dawkins
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.
Geometric functions in
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