Nonlinear Control Sy.. - Free

2.8. DIFFERENTIABILITY
2.8 Differentiability
Definition 2.20 A function f : R -> R is said to be differentiable at x if f is defined in an
open interval (a, b) C R and
exists.
f(x + hh - f(x)
f'(x) = l
a
The limit f'(x) is called the derivative of f at x. The function f is said to be
differentiable if it is differentiable at each x in its domain. If the derivative exists, then
f(x + h) - f(x) = f'(x)h + r(h)
where the "remainder" r(h) is small in the sense that
lim r(h) = 0.
h,O h
Now consider the case of a function f : R" -+ R.
Definition 2.21 A function f : R' -4 R' is said to be differentiable at a point x if f is
defined in an open set D C R" containing x and the limit
exists.
lim If(x+h) - f(x) - f'(x)hIl = 0
h-+0 lhll
Notice, of course, in Definition 2.21 that h E R'. If IhII is small enough, then
x + h E D, since D is open and f (x + h) E JRt.
The derivative f'(x) defined in Definition 2.21 is called the differential or the total
derivative of f at x, to distinguish it from the partial derivatives that we discuss next. In
the following discussion we denote by f2, 1 < i < 7n the components of the function f, and
by {el, e2, , en} the standard basis in R"
fi (x)
f2(x)
f(x) = I
Lfm(x)I
el =
fol f°1 fBl
,e2 =
0 0
en =
1
49