Calculus_Cheat_Sheet_Derivatives

Calculus_Cheat_Sheet_Derivatives - Calculus Cheat Sheet...

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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Derivatives Definition and Notation If ( ) y fx = then the derivative is defined to be () ( ) ( ) 0 lim h fx h h +- ¢ = . If ( ) = then all of the following are equivalent notations for the derivative. ( ) d f d yd f x y f x D d x d x dx ¢¢ === == If ( ) = all of the following are equivalent notations for derivative evaluated at xa = . x a d f dy f a y D fa d x dx = ==== Interpretation of the Derivative If ( ) = then, 1. ( ) m fa ¢ = is the slope of the tangent line to ( ) = at = and the equation of the tangent line at = is given by ( ) ( )( ) y f a faxa ¢ = . 2. ( ) ¢ is the instantaneous rate of change of ( ) at = . 3. If ( ) is the position of an object at time x then ( ) ¢ is the velocity of the object at = . Basic Properties and Formulas If ( ) and ( ) gx are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) () c f c ¢ ¢ = 2. ( ) () () f g fx gx ¢ =– 3. ( ) f g f g fg ¢ =+ – Product Rule 4. 2 f f g gg ¢ - = ²³ Ll – Quotient Rule 5. 0 d c dx = 6. ( ) 1 nn d x nx dx - = – Power Rule 7. d x f gxgx dx = This is the Chain Rule Common Derivatives 1 d x dx = ( ) si n cos d xx dx = ( ) co s sin d dx =- 2 ta n sec d dx = ( ) se c se c tan d x dx = ( ) cs c cs c cot d x dx 2 co t csc d dx 1 2 1 sin 1 d x dx x - = - 1 2 1 cos 1 d x dx x - - 1 2 1 tan 1 d x d - = + ln d aaa dx = ( ) d dx = ee 1 l n ,0 d d => ( ) 1 l n d d =„ 1 lo g ln a d d x
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Chain Rule Variants The chain rule applied to some specific functions. 1. () ( ) () () 1 nn d f x nf x fx dx - ¢ = Øø Øø ºû ºû 2. f x d dx ¢ = ee 3. ln d d x ¢ = Øø ºû 4. ( ) () () si n cos d f x f x dx ¢ = Ø ø º û 5. co s sin d f x f x dx ¢ =- Ø ø º û 6. 2 ta n sec d f x f x dx ¢ = Ø ø º û 7. [ ] ( ) [ ] [ ] ( ) ( ) ( ) se c se c tan f x f x f x d dx ¢ = 8. 1 2 tan 1 d dx - ¢ = + Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df f xfx dx ¢¢ == and is defined as f x ¢ ¢ = , i.e. the derivative of the first derivative, ¢ .
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This note was uploaded on 10/18/2008 for the course MATH 115 taught by Professor Riggs during the Spring '05 term at Cal Poly Pomona.

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Calculus_Cheat_Sheet_Derivatives - Calculus Cheat Sheet...

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