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DERIVATIVE RULES
There are some important rules to remember when computing the derivative of functions. The first is the sum rule:
(4) (af + bg)’ = af’ + bg
where a and b are real numbers, f and g are functions, and () is the derivative operator. The next is the product rule:
(5) (fg)’ = gf’ + fg
As a corollary to the product rule, the following is also true:
(6) (af)’ = af
where a is a constant. This rule follows from the product rule because the derivative of a constant is 0 (the slope of the function f(x) = a is 0 for all x). The next rule is the quotient rule:
(7) (f)/(g)’ = (gf’ − fg)/(g2)
Finally, there is the chain rule. The chain rule dictates how the derivative is determined when the function f is written in terms of another function: f = f(k) where k = k(x):
(8) f(x) = f(k)k(x)
DERIVATIVES OF BASIC FUNCTIONS
The derivatives of some common functions are listed below. Using these given derivatives and the derivative rules listed above, the derivatives of a great many more complicated functions can be determined. Derivatives of other basic functions are available in the appendices of most calculus texts.
  • Powers:
    (9) f(xn) = nxn − 1
  • Trigonometric functions:
    (10) f(sinx) = cosx
    (11) f(cosx) =  − sinx
    (12) f(tanx) = sec2x
  • Exponential and logarithmic functions:
    (13) f(ex) = ex
    (14) f(lnx) = (1)/(x),     x > 0
    (15) f(logax) = (1)/(xlna),     x > 0
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