Week 4 differentiation
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Remark: we will use the Binomial Theorem which says that for
The exponential function
Theorem 3.5 allows us to define new continuous functions by power series (with non-zero radius of convergence). Here is the most important example.
Definition: the exponential function and the number
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Proof. Apply the Ratio Test to the series
to find for all Since the power series is absolutely convergent for all (the radius of convergence is ). By Theorem 3.5, it follows that is a continuous function on all ofFigure 4.1: The double series used to prove
To prove
we compare two methods of summation of the double series see Figure 4.1. We havehence summation by rows gives
We now calculate the
th diagonal sum (multiplying and dividing by for emphasis):By the Binomial Theorem, the expression in brackets is the expansion of
Thus, summation by diagonals givesWe claim that the sum of all numbers in this double series does not depend on the method of summation, and so
We need to justify this claim.If both
and are non-negative, then all the numbers are non-negative. In this case, Proposition 2.3 guarantees that the sum, is independent of the method of summation, and soWithout the assumption that
are non-negative, we can show that the sum of all the absolute values in the table is finite:so by Claim 2.7, the sum
is still independent of the method of summation, and we still have □
Discussion of the
It also follows that, for
The law of the exponential also tells us that
Therefore,
Notation:
Definition of the natural logarithm function
We are going to introduce the inverse function to
Proposition 4.2: properties of
The function
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Proof. Observe that
In particular, is positive for positive Then implies that is positive for all and is indeed a function from toFor all
we have If then as observed above, so We have shown that is strictly increasing, hence injective.To show that
is surjective, let be arbitrary. If note that as shown above. Also The function is continuous, so by the Intermediate Value Theorem there exists such thatIf
then and by the above, for some We then have by the law of the exponential. Finally, if then We have proved that is surjective, and so it is bijective. □
We immediately deduce
Theorem 4.3: natural logarithm
There is a strictly increasing continuous bijection
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Sketch of proof.
is a bijection from to so it must have an inverse which we denote and call the natural logarithm function. Inverse means that andUsing the Inverse Function Theorem 1.2, we conclude that
is strictly increasing and continuous.By definition of
for all Set to get By the law of the exponential, this equals Yet and so the answer simplifies to We proved the logarithm law, □
Differentiation of functions: an informal introduction
We begin the second part of the course: the theory of differentiation.
To differentiate a “smooth” function
Figure 4.2: The secant passing through the points
We first present the idea informally (rigorous definitions are below). Fix a point
As
Why differentiate functions? It turns out that derivatives appear in powerful results which allow us to approximate functions by extremely good functions — polynomials — and to represent some functions as sums of infinite power series. But first, we build up theory to
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• differentiate basic functions, such as polynomials, rational functions, exponential, logarithm, trigonometric and inverse trigonometric functions;
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• use rules of differentiation, to find derivatives of new functions constructed from basic functions.
Definition of the derivative of at
We now start our rigorous treatment of differentiation.
Definition: open neighbourhood of the point
An open neighbourhood of
Definition: differentiable at
Let
exists. The value of this limit is the derivative of
Remark: for
Definition: differentiable on an open interval.
Remark: if
Notation:
If a function
There are functions whose derivatives can be computed by definition, i.e., by calculating the limit given in the definition of
Example: derivative of a constant function.
Given
Justification: by definition, the derivative at
Remark: Remember that the limit,
For example, the expression
To conclude: when calculating a limit
Example: derivative of the function
Justification: by definition, the derivative of
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Proof. The criterion of continuity says that
is continuous at iff Rearranging, we obtain: is continuous atAssume
is differentiable at so that the limit exists. Then Thus, verifies the (rearranged) criterion of continuity above, so is continuous at □
Alert: continuous at
The converse to Theorem 4.4 does not hold. For example,
Figure 4.3: Visibly, the graph of
Justification. “Differentiable at
The one-sided limits are not equal, so the limit
Rules of differentiation: sums and products
We can obtain new differentiable functions from known ones by addition and multiplication.
Theorem 4.5: sum and product rules of differentiation.
Suppose that the functions
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• the function
is differentiable at and -
• the function
is differentiable at and
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Proof. The sum rule (proof not given in class): by definition of the function
is the same as which rearranges as Taking the limit as and using AoL for functions, we obtain as claimed.The product rule: by definition,
Start withwhere we subtract then add
in the numerator. The RHS rearranges asWe are given that
is differentiable at Differentiable implies continuous, so is continuous at Hence Taking in the last displayed formula and using AoL, we get as claimed. □
Now, using only
Corollary.
A polynomial in
Differentiating infinite sums
The sum rule of differentiation does not extend to infinite sums. A function defined as a sum of series of differentiable functions may not be differentiable.
Yet one can show that a function defined as a sum of a power series is differentiable on
Let
We note that
One concludes from the above that
Instructions for the exam: differentiating a power series term-by-term as above without giving full justification will not be accepted in the exam. If asked to justify differentiation of
Proving “differentiable” by constructing slope function
Rather than showing directly that
Proposition 4.6: differentiability means continuity of the slope function at
A function
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Proof. If such
exists and is continuous at we have which, by continuity, is That is, exists and equalsNow suppose that
is differentiable at Then, definingguarantees
so by criterion of continuity is continuous at □
We call
This formula defines a polynomial function of
Differentiating
We use the method of continuous slope function to differentiate
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Proof. To differentiate
at writeThe slope function
is the sum of a power series convergent for all hence is continuous by Theorem 3.5, and by Proposition 4.6 exists and equals This proves the Special Limit forIndeed, the left-hand side is exactly the derivative of
at which we have just found to be We now differentiate at an arbitraryBy the Special Limit, this is
□
The Chain Rule and the Quotient Rule
We will work in the situation
We will write
Theorem 4.8: The Chain Rule.
If
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Proof. By Proposition 4.6, whenever
is differentiable at a point one hasfor all
where the slope function is continuous at In particular, this holds for andwhere we assumed that
was differentiable at and applied Proposition 4.6 toThe function
is continuous at because is continuous (even differentiable!) at is continuous at and a composition of continuous functions is continuous. The function is continuous at Therefore, by Algebra of Continuous Functions, is a continuous function of It immediately follows by Proposition 4.6 that the function is differentiable at withas its derivative at
as claimed. □
Example.
Find
Solution. Put
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•
is a polynomial, hence is differentiable for all with -
•
is differentiable for all by Proposition 4.7, with
Hence we are allowed to use the Chain Rule:
Corollary: the Quotient Rule.
If
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Proof. If
then, for any When calculating we may assume that so this simplifies toWriting
as and applying the Chain Rule, we have as claimed. Now, to obtain apply the Product Rule to □
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