Page 76 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 67

RIGHT- AND LEFT-HAND DERIVATIVES
The status of the derivative at end points of the domain of f ,and in other special circumstances, is
clarified by the following definitions.
The right-hand derivative of f ðxÞ at x ¼ x 0 is defined as

f þ ðx 0 Þ¼ lim f ðx 0 þ hÞ f ðx 0 Þ ð3Þ
0
h
h!0þ
if this limit exists. Note that in this case hð¼ xÞ is restricted only to positive values as it approaches
zero.
Similarly, the left-hand derivative of f ðxÞ at x ¼ x 0 is defined as

f ðx 0 Þ¼ lim f ðx 0 þ hÞ f ðx 0 Þ ð4Þ
0
h
h!0
if this limit exists. In this case h is restricted to negative values as it approaches zero.
A function f has a derivative at x ¼ x 0 if and only if f þ ðx 0 Þ¼ f ðx 0 Þ.
0
0
DIFFERENTIABILITY IN AN INTERVAL
If a function has a derivative at all points of an interval, it is said to be differentiable in the interval.
In particular if f is defined in the closed interval a @ x @ b, i.e. ½a; bŠ, then f is differentiable in the
interval if and only if f ðx 0 Þ exists for each x 0 such that a < x 0 < b and if f þ ðaÞ and f ðbÞ both exist.
0
0
0
If a function has a continuous derivative, it is sometimes called continuously differentiable.
PIECEWISE DIFFERENTIABILITY
A function is called piecewise differentiable or piecewise smooth in an interval a @ x @ b if f ðxÞ is
0
piecewise continuous. An example of a piecewise continuous function is shown graphically on Page 48.
An equation for the tangent line to the curve y ¼ f ðxÞ at the point where x ¼ x 0 is given by
0
y f ðx 0 Þ¼ f ðx 0 Þðx x 0 Þ ð7Þ
The fact that a function can be continuous at a point and yet not be differentiable there is shown
graphically in Fig. 4-3. In this case there are two tangent lines at P represented by PM and PN. The
slopes of these tangent lines are f ðx 0 Þ and f þ ðx 0 Þ respectively.
0
0

DIFFERENTIALS
Let x ¼ dx be an increment given to x. Then

y ¼ f ðx þ xÞ f ðxÞ ð8Þ
is called the increment in y ¼ f ðxÞ. If f ðxÞ is continuous and has a continuous first derivative in an
interval, then
y ¼ f ðxÞ x þ x ¼ f ðxÞdx þ dx ð9Þ
0
0
where ! 0as x ! 0. The expression
dy ¼ f ðxÞdx ð10Þ
0
is called the differential of y or f(x) or the principal part of y. Note that y 6¼ dy in general. However
if x ¼ dx is small, then dy is a close approximation of y (see Problem 11). The quantity dx, called the
differential of x, and dy need not be small.