Calculus_Cheat_Sheet_Derivatives_Reduced

Calculus_Cheat_Sheet_Derivatives_Reduced - Calculus Cheat...

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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 fxh f x fx h +− = . If yfx = then all of the following are equivalent notations for the derivative. df dy d f xy f x D f x dx dx dx ′′ == = = = If = all of the following are equivalent notations for derivative evaluated at x a = . xa df dy f ay D f a dx dx = == = = Interpretation of the Derivative If yf x = 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 ( ) y fa f a x a =+ . 2. ( ) f a is the instantaneous rate of change of ( ) f x at x a = . 3. If ( ) is the position of an object at time x then ( ) f a is the velocity of the object at x a = . Basic Properties and Formulas If f x and g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) c f c f x = 2. ( ) fg f x g x ±= ± 3. f gf g f g – Product Rule 4. 2 f fg fg gg ⎛⎞ = ⎜⎟ ⎝⎠ – Quotient Rule 5. 0 d c dx = 6. 1 nn d x nx dx = – Power Rule 7. d f gx f gx g x dx = This is the Chain Rule Common Derivatives 1 d x dx = sin cos d x x dx = cos sin d x x dx =− 2 tan sec d x x dx = sec sec tan d x xx dx = csc csc cot d x dx 2 cot csc d x x 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 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 d nfx dx = ⎡⎤ ⎣⎦ 2. ( ) f xf x d dx = 3. ln f x d dx f x = 4. sin cos d f x f x f x dx = 5. cos sin d f x f x f x dx 6. 2 tan sec d f x f x dx = 7. [] [][] sec sec tan f x f x f x d dx = 8. ( ) 1 2 tan 1 d dx f x = +⎡ Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df fx f x dx and is defined as f x = , i.e. the derivative of the first derivative, f x .

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Calculus_Cheat_Sheet_Derivatives_Reduced - Calculus Cheat...

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