文档介绍：Section
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Differentiation of Polynomials
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In this section we begin the task of discovering rules for differentiating various classes
of functions. By the end of Section we will be able to differentiate any algebraic or
trigonometric function as a matter of routine without reference to the limits used in Section
.
Differentiation of polynomials
We first note that if f is a first degree polynomial, say, f(x) = ax + b for some constants
a and b, then f is an affine function and hence its own best affine approximation. Thus
f 0(x) = a for all x. In particular, if f is a constant function, say, f(x) = b for all x, then
f 0(x) = 0 for all x.
Next we consider the case of a monomial f(x) = xn, where n is a positive integer
greater than 1. Then
f(x + h) − f(x) (x + h)n − xn
f 0(x) = lim = lim . ()
h→0 h h→0 h
Now
(x + h)n = xn + nxn−1h + R(h), ()
where R(h) represents the remaining terms in the expansion. Since every term in R(h)
has a factor of h raised to a power greater than or equal to 2, it follows that R(h) is o(h).
Hence we have
xn + nxn−1h + R(h) − xn
f 0(x) = lim
h→0 h
nxn−1h + R(h)
= lim
h→0 h
− R(h)
= lim nxn 1 +
h→0 h
R(h)
= nxn−1 + lim
h→0 h
= nxn−1.
Since from our previous result f 0(x) = 1 when f(x)