Calculus and CE - Calculus Cheat Sheet Derivatives...

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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 () yf x = then the derivative is defined to be ( ) ( ) 0 lim h fx h fx fx h +− = . If yfx = then all of the following are equivalent notations for the derivative. df dy d fx y D dx dx dx ′′ == = = = If ( ) = all of the following are equivalent notations for derivative evaluated at x a = . xa df dy fa y D fa dx dx = = = Interpretation of the Derivative If = then, 1. mfa = is the slope of the tangent line to = at x a = and the equation of the tangent line at x a = is given by () () ( ) yfa faxa =+ . 2. ( ) is the instantaneous rate of change of ( ) at x a = . 3. If ( ) is the position of an object at time x then is the velocity of the object at x a = . Basic Properties and Formulas If and gx are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) cf cf x = 2. ( ) () () fg fx gx ±= ± 3. fg f g fg – Product Rule 4. 2 ff g f g gg ⎛⎞ = ⎜⎟ ⎝⎠ – Quotient Rule 5. 0 d c dx = 6. 1 nn d xn x dx = – Power Rule 7. d fgx f gx gx dx = This is the Chain Rule Common Derivatives 1 d x dx = sin cos d xx dx = cos sin d dx =− 2 tan sec d dx = sec sec tan d x dx = csc csc cot d x dx 2 cot 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 dx x = + ln d aaa dx = d dx = ee 1 ln , 0 d dx x => 1 ln , 0 d dx x =≠ 1 log , 0 ln a d dx x a
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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 fx nfx f x dx = ⎡⎤ ⎣⎦ 2. d dx = ee 3. ln d dx f x = 4. sin cos d dx = 5. cos sin d dx =− 6. 2 tan sec d dx = 7. [] sec sec tan d dx = 8. 1 2 tan 1 d dx = +⎡ Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df fx f x dx ′′ == and is defined as = , i.e. the derivative of the first derivative, .
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This note was uploaded on 01/17/2011 for the course CE CE taught by Professor Armstrong during the Spring '10 term at The University of Texas at San Antonio- San Antonio.

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Calculus and CE - Calculus Cheat Sheet Derivatives...

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