MATH12002 - CALCULUS I
            §2.3: Basic Differentiation Formulas, Part 1
                         Professor Donald L. White
                        Department of Mathematical Sciences
                              Kent State University
   D.L. White (Kent State University)                                   1 / 6
  Differentiation Formulas
   There are two types of differentiation formulas:
        Formulas for derivatives of specific functions, such as xn, sinx, cosx,
        secx, lnx, etc.
        Formulas for computing new derivatives from old, such as the
        derivatives of f (x) + g(x), f (x) · g(x), f (x)/g(x), or f (g(x)) in
        terms of the derivatives of f (x) and g(x).
   Whatever the type, however, all of the formulas are based on the definition
   of derivative,
                          f ′(x) = lim f (x + h) − f (x),
                                  h→0         h
   and previously derived formulas.
   Wewill start with some derivatives of specific functions.
   D.L. White (Kent State University)                                        2 / 6
  Constant and Power Functions
   If c is a constant (i.e., a real number), then the graph of the constant
   function f (x) = c is simply a horizontal line.
   Its slope (and hence its derivative) is therefore always 0.
   Derivative of a Constant Function
   If f (x) = c for some constant c, then f ′(x) = 0; that is, d (c) = 0.
                                                             dx
   (This formula is also easy to prove using the definition directly.)
   In fact, if f (x) = mx + b is any linear function, then its slope, and hence
   derivative, is always m, and so we have
                                d (mx +b) = m
                                dx
   and, in particular,
                                    d (x) = 1.
                                   dx
   D.L. White (Kent State University)                                       3 / 6
  Constant and Power Functions
   Next, we consider the power function f (x) = xn.
   Using the Binomial Theorem (or direct calculation), we can compute
               1
        (x +h) =x +h
               2     2           2
        (x +h) =x +2xh+h
               3     3      2        2    3
        (x +h) =x +3x h+3xh +h
               4     4      3      2 2       3     4
        (x +h) =x +4x h+6x h +4xh +h
               5     5      4        3 2       2 3       4    5
        (x +h) =x +5x h+10x h +10x h +5xh +h
                                     n
   Using these expansions of (x + h) , it is easy to compute the derivative of
   f (x) = xn for n = 1,2,3,4,5. Do it!
   More generally, the Binomial Theorem says that if n is a positive integer,
   then
                  n    n      n−1                              2
          (x +h) =x +nx           h+[sum of terms with an h factor].
   D.L. White (Kent State University)                                        4 / 6