Key Ideas about Derivatives (3/20/09)

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Presentation transcript:

Key Ideas about Derivatives (3/20/09) The definition and meaning: f ‘ (x) = lim h -> 0 (f (x+h ) – f (x)) / h f ‘ (x) is the instantaneous rate of change of the function f at the point (x, f (x)). If f is graphed, f ‘ is the slope of the tangent line to the graph at that point.

The Rules The Rules tell us how to take the derivative of combinations of functions. Constant Multiplier Rule Sum and Difference Rule Product Rule Quotient Rule Chain Rule

The Facts The Facts tell us how to take the derivative for different classes of common functions. d/dx ( x r ) = r x r – 1 (r any constant) d/dx ( a x ) = a x ln(a) (a constant > 0) In particular, d/dx ( e x ) = e x d/dx ( loga (x ) = 1 / (x ln(a)) In particular, d/dx ( ln (x) ) = 1 / x

The Facts (Continued) d/dx (sin(x)) = cos(x) d/dx (cos(x)) = - sin(x) d/dx (tan(x)) = sec2(x) d/dx (arcsin(x)) = 1 /(1-x 2) d/dx (arccos(x)) = -1 /(1-x 2) d/dx (arctan(x)) = 1 / (1+x 2) Etc.