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Lecture

# Derivatives Cheat Sheet

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University of Ottawa

Mathematics

MAT1320

Kirill Zaynullin

Fall

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Calculus Cheat Sheet
Derivatives
Definition and Notation
f(x +h ) f x( )
If y = f( )then the derivative is defined to be ( )= lim .
hﬁ0 h
If y = f( )then all of the following are If y = f( )all of the following are equivalent
equivalent notations for the derivative. notations for derivative evaluated at x = a.
f¢( )= y = df = dy = d (f ( ))= Df x( ) f ( ) = y¢ = df = dy = Df ( )
dx dx dx x a dx x a dx x a
Interpretation of the Derivative
If y = f( )then, 2. f ( )s the instantaneous rate of
1. m = f a is the slope of the tangent change of f x at x = a.
( ) ( )
line toy = f ( )t x = aand the 3. If f ( )s the position of an object at
equation of the tangent line at= a is
time x then f ( )s the velocity of
given by y = f ( ) f a ( )(. ) the object at x = a.
Basic Properties and Formulas
If f( )and g x ( ) differentiable functions (the derivative exists), c and n are any real numbers,
¢ ¢ d
1. (c f) = c f ( ) 5. dx ( )= 0
¢ ¢ ¢ d
2. (f – g)= f ( ) – g ( ) 6. ( ) = nxn-1– Power Rule
3. f g = f g + f g – Product Rule dx
( ) d ¢ ¢
¢ 7. dx (f ( x( )))= f (g ( ))( )
4. ç f÷ = f g - f g¢ – Quotient Rule This is the Chain Rule
Łg ł g 2
Common Derivatives
d d d
( )=1 (cscx ) -cscxcot x ( ) = a ln ( )
dx dx dx
d d 2 d x x
dx(sin x)= cosx dx(cot x)= -csc x dx ( ) = e
d d 1 d 1
(cosx ) -sin x (sin x =) (ln( ))= , x > 0
dx dx 1- x 2 dx x
d tan x = sec x d -1 1 d ln x = 1 , x „ 0
dx( ) dx(cos x =)- 2 dx ( ) x
d 1- x d 1
(secx ) secxtan x d tan x = 1 (loga( ))= , x > 0
dx dx( ) 1+ x2 dx xlna
Visit http://tutorial.math.lamar.edute set of Calculus notes. © 2005 Paul Dawkins Calculus Cheat Sheet
Chain Rule Variants
The chain rule applied to some specific functions.
d n n-1 d
1. (º f ( )ß ) = nº f( )øß f ( ) 5. (cosØºf ( ) ß)- f ( ) sinº f ( )ß
dx dx
d f( ) f( ) d 2
2. dx ( ) = f¢( )e 6. dx (tanØºf ( ) ß)= f¢( )sec Øºf ( ) ß
3. d lnØ f ( ) = f¢( ) 7. d (sec [f (x])= f (xsec [f (x] [n f (x]
dx ( º ß ) f( ) dx
d f¢( )
4. d (sinØ f ( ) =)f ( ) cosØ f ( ) 8. (tan Ø º x( )ß ) 2
dx º ß º ß dx 1+ º f( )øß
Higher Order Derivativesth
The Second Derivative is denoted as The n Derivative is denoted as
( ) d f ( ) d f
f¢( )= f ( ) = 2 and is defined as f ( )= n and is defined as
dx dx
f¢( ) = ( f¢( ))¢, i.e. the derivative of the ( ) (n-) ¢
f ( )= f( ( ) ) , i.e. the derivative of
first derivative, ¢( ). the (n-1) derivative, f (-1) x .
( )
Implicit Differentiation
Find y if e 2x-9y + x y = sin ( )11x . Remember y = y x her( )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
differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).
After differentiating solve for y .
2x-9y 2 2 3
e (2-9 y ) +3 x y +2 x y y¢= cos ( )y ¢+11 2x-9y 2 2
2e 2x-y -9y e 2x-9y+3x y +2x y y = cos y y ( ) ¢ Þ y = 11-2e -3x y
2 x y -9e 2x-9y-cos ( )
(2 x y -9e 2x-y -cos ( ) )y ¢=11-2e 2x-9y -3 x y2
Increasing/Decreasing – Concave Up/Concave Down
Critical Points
x = c is a critical point of f x provided either Concave Up/Concave Down
( )
1. f c( )0 or 2. f c d( )n’t exist. 1. If f ¢¢( )> 0 for all x in an interval I then
f( )is concave up on the interval I.
Increasing/Decreasing 2. If f ¢¢( )< 0 for all x in an interval I then
1. If f ¢( )> 0 for all x in an interval I then
f( )is concave down on the interval I.
f( )is increasing on the interval I.
2. If f¢( )< 0 for all x in an interval I then Inflection Points
f( )is decreasing on the interval I. x = c is a inflection point of ( ) if the
concavity changes at x = c.
3. If f ¢( )= 0 for all x in an interval I then
f( )is constant on the interval I.
Visit http://tutorial.math.lamar.edulete set of Calculus notes. © 2005 Paul Dawkins Calculus Cheat Sheet
Extrema
Absolute Extrema Relative (local) Extrema
1. x = cis an absolute maximum of f x ( ) 1. x = c is a relative (or local) maximum of
f( )if f ( ) f x ( ) all x near c.
if f( )‡ f ( )or all x in the domain.
2. x = c is an absolute mini

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