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# Derivatives Cheat Sheet

4 Pages
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School
University of Ottawa
Department
Mathematics
Course
MAT1320
Professor
Kirill Zaynullin
Semester
Fall

Description
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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