Differentiation
The slope of a line is a fundamental concept. The notion of a derivative is a generalization
of slope of a line to “slope of a curve at a specific point.”
Some modern calculus texts are starting to refer to differentiable functions as locally linear
functions. This characterization of differentiability is a good one for calculus students! Informally,
we say that the function y = f(x) is locally linear at x = a provided that when you zoom-in on the
curve segment at (a,f(a)) it resembles a line segment. The slope of this “line segment” is the value
of the derivative at x = a. This notion is made formal in the Fundamental Lemma of
Differentiation.
Example 29
Consider the graphic below showing the graph of
the point
in three viewing windows near
. The rightmost graphic below appears to show the graph of a function that is
nearly linear and the “slope” of that “line segment” is roughly
(Of course, by basic calculus,
.
.)
Definition 42
Let f be a real-valued function defined on a nontrivial interval J and let c 0 J. We say that f is
differentiable at x = c iff
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exists (and is finite). (If J = [a,b] and if c = a or c = b, the above two-sided limit is replaced by
the appropriate one-sided limit.) In the case the limit exists, we write
and call
the derivative of f at x = c.
We regard
as a function with
Another way to compute
and rule of assignment
.
(provided it exists) is given by
.
We refer to either
(*)
or
(**)
as the difference quotient for f at x = c. (We note that (*) and (**) are sometimes called the Newton
quotient for f at x = c.)
The line given by
is called the tangent line for the graph of
at points where
given by
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exists. Further, the line
is called the normal line for the graph of
at points where
exists and is not zero.
Example 30
1.
Let k 0 ú. If
2.
Let n 0 ù. If
3.
If
on an interval J, then
on J.
on an interval J, then for all
, then for all real x we have that
Note:
4.
If
, then for all
we have that
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we have that
5.
Define a function f on ú by
We consider the difference quotient for f at x = 0:
Since
and so, by the Sandwiching Theorem,
,
. (Of course, for
we have that
Example 31
1.
Show that the function
fails to have a derivative at the origin.
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.)
Solution
Consider
We conclude that
does not exist as a finite real number. We note that the graph of
has a vertical tangent line at the origin.
2.
Show that
fails to have a derivative at the origin.
Solution
Consider
which fails to exists at x = 0. We conclude that
note that the graph of
does not exist as a finite real number. We
has a “corner” at the origin.
Theorem 43 - Sequential Criterion for Differentiability
Suppose that f is defined on a nontrivial interval J. Then f is differentiable at x = c iff there exists
a real number m such that for every sequence
.
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with
, the sequence
Example 32
Define f on ú by
We note that f is continuous on ú. Set
and
. Then
.
However,
and
.
So, by Theorem 43, we conclude that the derivative of f at x = 0 does not exist. The graphic below
illustrates the “nonlinear” behavior of
near x = 0:
We now show that the class of differentiable functions is a subset of the class of continuous
function.
Theorem 44
If f is differentiable at x = c, then f is continuous at x = c.
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Proof
We assume that f has a two-sided derivative at x = c. Otherwise, replace all two-sided limits that
follow with the appropriate one-sided limit. For x … c,
.
Therefore,
We conclude that
and so f is continuous at x = c. >
Of course,
and
are both continuous at x = 0 yet neither is differentiable there.
So, the class of differentiable functions is a (proper) subset of the class of continuous functions.
Example 33
Prove or disprove: The function f defined by
is differentiable at x = 0.
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Solution
It might be tempting to say that
(since
) but it
turns out that f is not continuous at x = 0 and so, by Theorem 44, f can not be differentiable there.
The converse of Theorem 43 is not true. In 1872, Karl Weierstrass proved the existence of
a function that is everywhere continuous yet nowhere differentiable.
Example 34
1. (Weierstrass) Define f on ú as follows:
.
Then f is everywhere continuous yet nowhere differentiable. The graph of
is shown
below:
2. Let
be defined by
for
and let
. (That is, f is a periodic (p = 1) sawtooth function.) We now define
.
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for all
by
(We note that the graph of
consists of a sequence of line segments with slope
(i.e., sawtooth function) and period equal to
.) One can prove by the Weierstrass
M-Test that the above series converges uniformly on ú and, hence, is continuous there. Of course,
g is periodic on ú with period 1. The graph of g is shown below:
The graphics below shows successive zooms for the graph of g near
:
We now show that g is nowhere differentiable. Fix x in ú. For each n choose
and
. If
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so that
then, it follows that
and so
where
. Thus,
is an odd integer if n is even
and an even integer when n is odd. (Easy induction proof!) It now follows from Theorem 43 that
g is not differentiable at x. Since x was arbitrary, f is nowhere differentiable. >
How many functions like those of Example 34 are there? The next result shows that “near” every
continuous function there is another continuous function that is nowhere differentiable.
Theorem 45
Let
h:[0,1]6ú be continuous.
For each
there exists a function
g:[0,1]6ú with
for all x in [0,1] and such that g is continuous and nowhere differentiable.
As the above theorem should suggest, the continuous, everywhere differentiable functions
encountered in Calculus I are really the exceptional continuous functions and are not the norm!
Further, every differentiable (and, hence, continuous) function is the uniform limit of continuous yet
nowhere differentiable functions.
We now state the usual differentiation rules (sum, difference, product, & quotient).
Theorem 46
Suppose that f and g are defined on [a,b] and are differentiable at x 0 [a,b]. Then the functions
,
, and
are differentiable at x with
(i)
(ii)
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(iii)
.
Proof (Sketch)
(i) Clear.
(ii) Note:
Letting t 6 x we obtain the product rule.
(iii) Observe:
Letting t 6 x we obtain the quotient rule. >
We next result illustrates why modern calculus text refer to differentiable functions as locally
linear functions.
Theorem 47 - Fundamental Lemma of Differentiation
Suppose that f is differentiable at the interior point x = c. Then there exists a function
on the interval
(some ä > 0) so that
with
.
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defined
Proof
Solving
for
we obtain
.
At the exceptional point we define
although this is not required by the conclusion of the
theorem. Since f is differentiable at x = c,
or, equivalently,
.
From the above we conclude that
. (The “error” function
For x near c we have that
is in fact continuous at x = c.) >
and so
is “extremely” close to zero (why?). It
follows that
.
The above is the tangent line approximation to y = f(x) with base point x = c. The figure below
shows an arbitrary differentiable function, a typical tangent line approximation and the “error”
function
.
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Example 35
Show that if f is differentiable at x = c, then
.
Solution
By the Fundamental Lemma of Differentiation we have
and
.
Substitution of (*) and (**) into
yields
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or
.
Letting h 6 0 in the above we obtain the desired result since
. >
Observe that when á + â = 1 the above returns the derivative of f at x = c. It is important to note
that the limit
(some á & â) may exist even though the function f fails to have a derivative at x = c. For example,
take f(x) = |x|, á = â = 1, and c = 0.
We now use the Fundamental Lemma of Differentiation to prove the Chain Rule.
Theorem 48 - Chain Rule
If g is differentiable at x = c and f is differentiable at x = g(c), then f B g is differentiable at x =
c and
.
Proof
By the Fundamental Lemma of Differentiation applied to f at g(c) we have
with
.
Setting z = g(x) we may rewrite the above as
.
Because g is differentiable and, hence, continuous at x = c,
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and
.
Letting x 6 c in (**) and using the above two facts we obtain the Chain Rule. >
Example 36
Show that
.
Solution
Let
. Then cos y = x. By the chain rule,
and so
.
By basic right triangle trigonometry,
Thus,
.
The above serves as a model for the next theorem.
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Theorem 49 - Inverse Differentiation Theorem
Suppose that f is a continuous, (strictly) increasing function which has an interval I for domain and
has J for range.
(1)
The inverse function
is continuous and increasing on J;
(2)
If
exists and is not zero, then
such that
exists and
.
The above also is valid with “decreasing” replacing “increasing.” The proof of Theorem 49 is similar
to that of Example 36. (Observe that
when
.)
We assume the usual definitions of local / relative maximum and local / relative minimum
of a function on an interval from ordinary calculus.
Theorem 50
and suppose f:I6ú has a relative extrema at
Let
, then
. If f is differentiable at
.
Proof
We assume that f has a local maximum at
manner.) On the contrary, suppose that
. (The local minimum case is done a similar
.
Case 1
Assume that
. Since
,
there exists ä > 0 such that for
we have
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.
So, for
we have
and so
.
A contradiction of the fact that f has a local maximum at
.
Case 2
Assume that
. ... A contradiction.
Hence, if f is differentiable at
, then
. >
The following is an immediate corollary to the above theorem.
Corollary 51
and suppose f : I 6 ú has a relative extrema at
Let
f fails to be differentiable at
. Then either
.
From our experience in calculus we tend to think that if the differentiable function
local minimum at, say,
the right of
or
, then f is decreasing just to the left of
has a
and increasing just to
. The next example serves to show that this thinking has its shortcomings.
Example 37
Let f be defined by
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.
Then one can establish that f has an absolute minimum at the origin. The derivative of f is given
by
and, in particular, we see that
. However,
assumes both positive and negative
values in every open interval containing the origin. The graphic below shows the graphs of
and
in a (small) neighborhood of the origin:
Recall that a point
differentiable at
is called a critical point of f iff either
or f fails to be
. Hence, relative extrema for a function f occur at the “boundary points”
of the domain, if any, or at the critical points, if any.
The Mean Value Theorem (or Law of the Mean) due to Joseph LaGrange is one of the
cornerstone results in differential calculus. We establish the so-called Cauchy’s Generalized Mean
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Value Theorem, which has the Mean Value Theorem of ordinary calculus as one of its
consequences.
Theorem 52 - Cauchy’s Generalized Mean Value Theorem
Suppose that f and g are two functions that are
(1)
continuous on
(2)
differentiable on
;
.
Then there exists a number
so that
.
Proof
Define h:[a,b]6ú by
.
Then the function h is continuous on
and differentiable on
. If
some k,
then choose c to be any number between a and b and the desired result follows. So, assume
is not a constant function. By the Extreme Value Theorem,
maximum value and absolute minimum value on
, it follows that
50,
. Since
assume both its absolute
is not a constant function and
has at least one interior extrema, say at
. By Theorem
and the desired conclusion that
follows. >
Setting
in Cauchy’s Generalized Mean Value Theorem produces the classical
Mean Value Theorem of ordinary calculus.
Corollary 53 - Mean Value Theorem
If f is continuous on
and if f is differentiable on
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, then there exists a real number
so that
or
.
Geometrically the Mean Value Theorem says there is at least one point on the smooth curve
between
and
secant line through the two points
at which the line tangent to the curve is parallel to the
and
. The following graphic illustrates the
MVT.
Corollary 54 - L’Hospital’s Rule for 0/0
Suppose that f and g are continuous functions on
. If
and
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and that
and
exist on I with
,
then
.
Proof
Define f and g at a by
. Since
and
, by the MVT we
have that
and so
. Let
. Because
,
there exists a
for all
so that
. Since
,
.
Applying Cauchy’s Generalized MVT on
, there exists
.
Hence,
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so that
.
Since
was arbitrary,
.>
The above extends to left-hand limits and then to two-sided limits giving us the classic L’Hospital’s
Rule.
Example 38
Show that for x > 0 that
.
Solution
for x in [0,4). Direct computation yields
Set
.
Fix x > 0. By the Mean Value Theorem there exists a real number c between 0 and x so that
or, more specifically,
.
Because x > 0 and
,
.
Thus,
and the desired result follows.
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Theorem 55
Let I be an interval and let f be differentiable on I. Then
(1)
f is strictly increasing on I iff
(2)
f is constant on I iff
(3)
f is strictly decreasing on I iff
for all x in I.
for all x in I.
for all x in I.
Proof (Sketch)
Let
. Applying the Mean Value Theorem to f on
with
for some
. Since
with
we obtain
, (1) - (3) follow from (*) >
Recall the Intermediate Value Theorem: If f is continuous on
number between
and
and
, then there exists at least one number
is any
so that
. The following result due to Duhamel illustrates that even though a derivative may fail
to be continuous at many points, it is a fact that derivatives satisfies the “intermediate value
property.”
Theorem 56
Suppose that f is differentiable on
and
with
, then there exists at least one number
. If
so that
Proof
Let
and define two functions on
as follows:
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is any number between
.
and
.
It follows that
are continuous on
with
. Further, we have that
.
Define a function g on
as follows:
.
Since g is the “composition” of continuous functions, g is itself continuous on
continuous at the endpoints of
. That g is
follows from the fact that
and, similarly,
.
So, g is seen to be a continuous function on
with
the Intermediate Value Theorem, there exists a number
Mean Value Theorem to f on the
yields
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a number between
such that
. By
. Applying the
for some
. >
Example 39
Show there exist functions f so that there is no function g with
.
Solution
. Then f has no antiderivative or primative since Theorem 56 assures us
Set
all derivatives satisfy the intermediate value property.
In fact, any function with a jump
discontinuity will not have an antiderivative.
An important task in analysis is the approximation of a function by polynomials. A major
theorem in this area is due to Brook Taylor (1685-1731) and usually bears his name. The result may
be viewed as an extension of the Mean Value Theorem to higher order derivatives.
Before we state and prove Taylor’s Theorem, we recall some standard notation from ordinary
calculus. Let f be a function. The derivative of f is denoted by
its derivative is denoted by
. If
is itself differentiable, then
and is called the second derivative of f. Continuing in this way, we
get a sequence of functions
each of which is the derivative of the preceding one. We call
that for
to exist at x that
we sometimes write
the nth derivative of f. We note
must exist on an open interval containing x. (For convenience,
,
, and
.)
By the Fundamental Lemma of Differentiation, if f is differentiable at
degree polynomial given by
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, then the first
for x near
is a good approximation to
. We observe that
and
.
Proceeding one step further, it would seem reasonable that a better approximation to f would have,
in addition to the properties of
, a common “curvature” at
. Such a function is given by
.
It is easy to verify that
,
,
and
.
The figure below shows the graphs of
function
at
and
.
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for the
The table below compares values of
,
for values of x near 0.
, and
x
-0.5
-0.4
-0.3
-0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
f(x)
0.6065
0.6703
0.7408
0.8187
0.9048
1.0
1.1051
1.2214
1.3499
1.4918
1.6487
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.625
0.68
0.745
0.82
0.905
1.0
1.105
1.22
1.345
1.48
1.625
As might be expected, the second degree polynomial
= 0 than the linear function
better near x
approximates
. Taylor’s Theorem extends this notion to higher order
polynomials.
Before stating Taylor’s Theorem, we note there exist other polynomial approximations to
“near” x = 0 meeting other reasonable criteria. For example, the least squares
approximation
for
relative
to
the
data
points
is given by
.
Theorem 57 - Taylor’s Theorem (with Lagrange’s Remainder)
Suppose that n 0 ù, that f and its derivatives
and that
exists on
are defined and continuous on
, then there exists a number c between
. If
and x so that
.
Proof - Time Permitting
Let
be fixed. Let
be the real number satisfying the equation
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.
Show that
Define a function è:J6ú by
.
The function è is continuous on
and differentiable on
Theorem, there exists a real number c between
and x so that
.
Since
,
.
Now,
Because
>
, the above implies that
We call the polynomial
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. By the Mean Value
the mth degree Taylor polynomial for the function y = f(x) centered at
.
Example 40
Suppose that
. (That is, suppose that
are all continuous on
.)
Show that
for all
.
Solution
Let
and let h be a real number such that
exist real numbers c and d in
. By Taylor’s Theorem, there
so that
and
.
(We note that both c and d depend on h as well as
.) Adding the above two equations together
and rewriting we obtain
.
Because
is continuous on
, the expression
. Hence, there exists M > 0 so that
.
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is bounded for
Letting h 6 0 produces the desired result. The result is true assuming only
exists at
. Clearly,
a different proof is required in this case. Try it!
The value of the above example is that for sufficiently nice functions f one can estimate
of f rather than
in terms
. That is, for small values of h we have
.
Example 41
Find n so that the nth degree Taylor polynomial centered at x = 0 that satisfies
for all x in [-ð,ð].
Solution
By Taylor’s Theorem, we have
We seek n so that
.
Trial and error shows that n = 20 is the smallest n satisfying (*). Hence,
approximate
on an eight digit calculator in the interval [-ð,ð].
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is good enough to