1. 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.

2. 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

3. 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

4. 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.