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Let’s now apply this definition to calculate a differentiation formula for a x . We have

d d x a x = d d x e x ln a = e x ln a ln a = a x ln a .

The corresponding integration formula follows immediately.

Derivatives and integrals involving general exponential functions

Let a > 0 . Then,

d d x a x = a x ln a

and

a x d x = 1 ln a a x + C .

If a 1 , then the function a x is one-to-one and has a well-defined inverse. Its inverse is denoted by log a x . Then,

y = log a x if and only if x = a y .

Note that general logarithm functions can be written in terms of the natural logarithm. Let y = log a x . Then, x = a y . Taking the natural logarithm of both sides of this second equation, we get

ln x = ln ( a y ) ln x = y ln a y = ln x ln a log a x = ln x ln a .

Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base a . Again, let y = log a x . Then,

d y d x = d d x ( log a x ) = d d x ( ln x ln a ) = ( 1 ln a ) d d x ( ln x ) = 1 ln a · 1 x = 1 x ln a .

Derivatives of general logarithm functions

Let a > 0 . Then,

d d x log a x = 1 x ln a .

Calculating derivatives of general exponential and logarithm functions

Evaluate the following derivatives:

  1. d d t ( 4 t · 2 t 2 )
  2. d d x log 8 ( 7 x 2 + 4 )

We need to apply the chain rule as necessary.

  1. d d t ( 4 t · 2 t 2 ) = d d t ( 2 2 t · 2 t 2 ) = d d t ( 2 2 t + t 2 ) = 2 2 t + t 2 ln ( 2 ) ( 2 + 2 t )
  2. d d x log 8 ( 7 x 2 + 4 ) = 1 ( 7 x 2 + 4 ) ( ln 8 ) ( 14 x )
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Evaluate the following derivatives:

  1. d d t 4 t 4
  2. d d x log 3 ( x 2 + 1 )
  1. d d t 4 t 4 = 4 t 4 ( ln 4 ) ( 4 t 3 )
  2. d d x log 3 ( x 2 + 1 ) = x ( ln 3 ) ( x 2 + 1 )
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Integrating general exponential functions

Evaluate the following integral: 3 2 3 x d x .

Use u -substitution and let u = −3 x . Then d u = −3 d x and we have

3 2 3 x d x = 3 · 2 −3 x d x = 2 u d u = 1 ln 2 2 u + C = 1 ln 2 2 −3 x + C .
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Evaluate the following integral: x 2 2 x 3 d x .

x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C

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Key concepts

  • The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
  • The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
  • The function e x is then defined as the inverse of the natural logarithm.
  • General exponential functions are defined in terms of e x , and the corresponding inverse functions are general logarithms.
  • Familiar properties of logarithms and exponents still hold in this more rigorous context.

Key equations

  • Natural logarithm function
  • ln x = 1 x 1 t d t Z
  • Exponential function y = e x
  • ln y = ln ( e x ) = x Z

For the following exercises, find the derivative d y d x .

y = 1 ln x

1 x ( ln x ) 2

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For the following exercises, find the indefinite integral.

d x 1 + x

ln ( x + 1 ) + C

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For the following exercises, find the derivative d y / d x . (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] y = x ln ( x )

ln ( x ) + 1

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[T] y = ln ( sin x )

cot ( x )

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[T] y = 7 ln ( 4 x )

7 x

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[T] y = ln ( ( 4 x ) 7 )

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[T] y = ln ( tan x )

csc ( x ) sec x

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[T] y = ln ( tan ( 3 x ) )

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[T] y = ln ( cos 2 x )

−2 tan x

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For the following exercises, find the definite or indefinite integral.

0 1 d t 3 + 2 t

1 2 ln ( 5 3 )

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0 2 x 3 d x x 2 + 1

2 1 2 ln ( 5 )

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2 e d x ( x ln ( x ) ) 2

1 ln ( 2 ) 1

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0 π / 4 tan x d x

1 2 ln ( 2 )

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( ln x ) 2 d x x

1 3 ( ln x ) 3

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For the following exercises, compute d y / d x by differentiating ln y .

y = x 2 + 1 x 2 1

2 x 3 x 2 + 1 x 2 1

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y = x −1 / x

x −2 ( 1 / x ) ( ln x 1 )

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y = e ln x

1 x 2

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For the following exercises, evaluate by any method.

5 10 d t t 5 x 10 x d t t

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1 e π d x x + −2 −1 d x x

π ln ( 2 )

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d d x x x 2 d t t

1 x

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d d x ln ( sec x + tan x )

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For the following exercises, use the function ln x . If you are unable to find intersection points analytically, use a calculator.

Find the area of the region enclosed by x = 1 and y = 5 above y = ln x .

e 5 6 units 2

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[T] Find the arc length of ln x from x = 1 to x = 2 .

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Find the area between ln x and the x -axis from x = 1 to x = 2 .

ln ( 4 ) 1 units 2

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Find the volume of the shape created when rotating this curve from x = 1 to x = 2 around the x -axis, as pictured here.

This figure is a surface. It has been generated by revolving the curve ln x about the x-axis. The surface is inside of a cube showing it is 3-dimensinal.
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[T] Find the surface area of the shape created when rotating the curve in the previous exercise from x = 1 to x = 2 around the x -axis.

2.8656

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If you are unable to find intersection points analytically in the following exercises, use a calculator.

Find the area of the hyperbolic quarter-circle enclosed by x = 2 and y = 2 above y = 1 / x .

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[T] Find the arc length of y = 1 / x from x = 1 to x = 4 .

3.1502

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Find the area under y = 1 / x and above the x -axis from x = 1 to x = 4 .

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For the following exercises, verify the derivatives and antiderivatives.

d d x ln ( x + x 2 + 1 ) = 1 1 + x 2

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d d x ln ( x a x + a ) = 2 a ( x 2 a 2 )

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d d x ln ( 1 + 1 x 2 x ) = 1 x 1 x 2

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d d x ln ( x + x 2 a 2 ) = 1 x 2 a 2

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d x x ln ( x ) ln ( ln x ) = ln ( ln ( ln x ) ) + C

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
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no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
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Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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I only see partial conversation and what's the question here!
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what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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What is meant by 'nano scale'?
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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