gaurav chaurasia(08613g)

8/3/2019 gaurav chaurasia(08613g)

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2011

JAYPEE UNIVERSITY OF

ENGINEERING AND

TECHNOLOGY, GUNA

GAURAV CHAURASIA

(08613g)

[DIFFERENTIATION OF EXPONENTIAL

FUNCTION]No matter how many times you differentiate e^x , it remains e^x. Exponential

functions only hold this property. But its not true for bases other than e.

Diffentiation of a^x is not a^x. This unique property of the function e^x can be

proved in several ways. Some of them are explained here which require the

knowledge of basic calculus.


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INTRODUCTION

The exponential function Y Xis the great creation of calculus.

Calculus is aboutpairs of functions. Function 1 (the distance we travel or the

height we climb) is changing. Function 2 (the velocity

or the slope

) tells the

rate of change. From one of those functions, we find the other.

This is the heart of calculus. The relation of Function 1 to Function 2 is learned byexamples more than by definitions, and those great functions are the right ones to

remember .

JIJ I

With as our goal, let me suggest that we go straight there. If we hide its bestproperty, students wont find it (and wont feel it). What makes this function

special ?

The slope ofXis X.Function 1 equals Function 2.

solves the differential equationY

Y

Restated, the function is equal to its own derivative, and satisfies the equationY

Y

is the solution ofY

Y that starts from y=1 atx=0.

Before that solution, draw what it means to have

The slope at x=0 must

be

(since y =1). So the curve starts upward, along the line y=1+x.

But as increases, its slope increases. So the graph goes up faster (and then faster).


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Exponential growth means that the function and its slope stay

proportional.

CHECKING THE VALIDITY OFY

YFOR Y X

APPROACH. 1 Constructing Y X

I will solve

a step at a time. At the start, y=1 means that

Start Change y Change

y=1 - -

-

After the first change, - has the correct derivative

. But then I have

to change

to keep it equal to y.

And I cant stop there:

y 1 - - -

cubic

equals 7 7 7 7

Y

1 - - -

cubic


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The extra

$gives the correct in the slope. Then

$also has to go into

, keep

it equal to y. now we need a new term with this derivative

$

The term that gives

$ has % divided by 6. The derivative of is # ,

so I must divide by n(to cancel correctly). Then the derivative of

is

%

$ as we wanted. After that comes & divided by 24.

{%{${#has slope

{${#

$&

{&{%{${#has slope

$&

&

{&{%{${#

The pattern becomes more clear. The term is divided by n factorial

which is {{ . { . {. The first five factorials are

1,2,6,24,120. The derivative of that term

is the previous term

{#

(because the ns cancel). As long as we dont stop, this sum of infinitely

many terms does achieve

.

Y{ X - -

-

- -

-

Here is the function. Take the derivative of every term and this series

appears again.

If we substitute x=10 into this series, do the infinitely many terms add to a

finite number #" ? Yes. The numbers grow much faster than (or any

other ). So the terms#"

in this exponential series become extremely


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small as . Analysis shows that the sum of the series (which is )

does achieve

Note:- Here is another way to look at that series for . Start with and

take its derivative n times. First get #and then { . $Finally

the nth derivative is { . { . {; which is nfactorial. When

we divide by that number, the nthderivative of

is equal to 1. All other

derivatives are zero at

Now look at

All its derivatives are still

; so they also equal 1 at

The series is matching every derivative ofXat the starting point

Here is the graph(figure.1) that shows-

Function(1)=Function(2)=X


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Figure.1

APPROACH. 2 Differentiation of a function f(x)

Recall that to differentiate any function, f(x), from first principles we find the slope

of theline joining an arbitrary point, A and a neighbouring point, B, on the

graph off(x). We then determine what happens to in the

limit as tends to

zero. (See Figure.2).


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Figure.2

The derivative,{, is then given by

{ =Y

{{

Lets consider the derivative of the exponential function going back to our limit

definition of the derivative.

We wish to find and use derivatives for functions of the form{ , where is a constant. By far the most convenient such function for this purpose is the

exponential function with base the special numberX.


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Definition. The numberX, which is approximately 2.7182818284590... , is thenumber such that

X

= 1

The numberX is called Euler's number, after the great mathematician LeonardEuler (1707-1783).

The major reason for the use ofX is the following theorem, which says thatX isits own derivative.

Theorem.{X

X

Proof. For any function {,

{="{{

Hence{

"

= "

= "{#

= { (by the definition)

=

This approach is basic one which is the tool for getting the differentiation of any

function. So this approach simply gives the result for{


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APPROACH. 3 USING KEY POINT

{ X{

To differentiate

we will rewrite this expression in its alternative formusing logarithms:

Then differentiating both sides with respect to ,{

The idea is now to find

.

We know that {

{

0

(This result is obtained using a technique known as the chain rule).

Now we know, that{

#

and so#

0

Rearranging,

But y = and so we have the important and well-known result that

The exponential function (and multiples of it) is the only functionwhich is equal to its derivative.


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CONCLUSION

The derivative of the Natural Exponential Function is itself i.e.

{X

X

This is the unique property of Natural Exponential Function which is true inevery case no matter how many times one differentiate this function.

This property shows that slope of Natural Exponential Function is equal tothe function itself at any point.

The above result makes the chain rule simpler if this function is multiplied toanother function.

This property also shows that Natural Exponential Function is alwaysmonotonic (without any break point). In particular, is strictly increasingand convex.

REFERENCES

1. Joh. Bernoulli, Principia calculi exponentialium, Opera I (1697) 179187.2. L.Euler,Introductio in analysin infinitorum, Opera 8, (1748).

gaurav chaurasia(08613g)

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