12 Differentiation
Differentiation
In physical terms, differentiation expresses the rate at which a quantity, y,
changes with respect to the change in another quantity, x, on which it has
a functional relationship. This small chapter will start with the discussion
of the derivative, which is one of the two central concepts of calculus (the
other is the integral). We will discuss the Mean Value Theorem and look at
some applications that include the relationship of the derivative of a function
with whether the function is increasing or decreasing. We will expose Taylor's
theorem as a generalization of the Mean Value Theorem. In calculus, Taylor's
theorem gives the approximation of a differentiable function near a point by a
polynomial whose coefficients depend only on the derivatives of the function
at that point. There are many OR applications of Taylor's approximation,
especially in linear and non-linear optimization.12.1 Derivatives
Definition 12.1.1 Let f : [a,b] K-> K. VX G [a, b], let (j>(t) = /(t)lf(x), a <
t < b, t ^ x. f'{x) = limt_>.x <j>(t) provided that the limit exists. f is called
the derivative of f. If f is defined at x, we say f is differentiable at x. If f
is defined at\fx € E C [a,b], we say f is differentiable on E. Moreover, left-
hand (right-hand) limits give rise to the definition of left-hand (right-hand)
derivatives.Remark 12.1.2 If f is defined on (a,b) and if a < x < b, then f can be
defined as above. However, f'(a) and f'(b) are not defined in general.
Theorem 12.1.3 Let f be defined on [a,b], f is differentiable at x G [a, b]
then f is continuous at x.
Proof. Ast-^x, f(t) - f{x) = f(t)tZfx{x) (t - x) -> f'{x) -0 = 0. •