classes_winter09_113AID28_Handout_2_Chem113A_W09

classes_winter09_113AID28_Handout_2_Chem113A_W09 -...

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Differential Calculus Differentiation of a Function Given some function f (x), the first derivative is defined as: h (x) - h) (x lim (x) dx d 0 h f f f + = What does this mean? For f (x)=x 2 , we find: 2x h) (2x lim h h 2xh lim h x - h 2xh x lim h (x) - h) (x lim x dx d 0 h 2 0 h 2 2 2 0 h 2 2 0 h 2 = + = + = + + = + = Continuing in this manner for other powers, we find the following differentiation power function rule: number some n for , n x ) (x dx d 1 - n n = = f Problem 1 : (a) Prove the power law differentiation rule is true for the following power function: f (x) = x 5 . (b) Using the formal definition of the derivative, find the first derivative for f (x) = ax, where a is some constant. Other useful derivatives -sin x x) (cos dx d x cos (sin x) dx d x 1 (ln x) dx d constant some is a where , ae ) (e dx d ax ax = = = = Quotient Rule (x) dx d (x)] [ 1 - (x) 1 dx d 2 f f f = Product Rule (x)] dx d (x) [ (x)] dx d (x) [ (x)] (x) [ dx d f g g f g f
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classes_winter09_113AID28_Handout_2_Chem113A_W09 -...

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