<< Chapter < Page Chapter >> Page >

Calculating derivatives of natural logarithms

Calculate the following derivatives:

  1. d d x ln ( 5 x 3 2 )
  2. d d x ( ln ( 3 x ) ) 2

We need to apply the chain rule in both cases.

  1. d d x ln ( 5 x 3 2 ) = 15 x 2 5 x 3 2
  2. d d x ( ln ( 3 x ) ) 2 = 2 ( ln ( 3 x ) ) · 3 3 x = 2 ( ln ( 3 x ) ) x
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Calculate the following derivatives:

  1. d d x ln ( 2 x 2 + x )
  2. d d x ( ln ( x 3 ) ) 2
  1. d d x ln ( 2 x 2 + x ) = 4 x + 1 2 x 2 + x
  2. d d x ( ln ( x 3 ) ) 2 = 6 ln ( x 3 ) x
Got questions? Get instant answers now!

Note that if we use the absolute value function and create a new function ln | x | , we can extend the domain of the natural logarithm to include x < 0 . Then ( d / ( d x ) ) ln | x | = 1 / x . This gives rise to the familiar integration formula.

Integral of (1/ u ) du

The natural logarithm is the antiderivative of the function f ( u ) = 1 / u :

1 u d u = ln | u | + C .

Calculating integrals involving natural logarithms

Calculate the integral x x 2 + 4 d x .

Using u -substitution, let u = x 2 + 4 . Then d u = 2 x d x and we have

x x 2 + 4 d x = 1 2 1 u d u 1 2 ln | u | + C = 1 2 ln | x 2 + 4 | + C = 1 2 ln ( x 2 + 4 ) + C .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Calculate the integral x 2 x 3 + 6 d x .

x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C

Got questions? Get instant answers now!

Although we have called our function a “logarithm,” we have not actually proved that any of the properties of logarithms hold for this function. We do so here.

Properties of the natural logarithm

If a , b > 0 and r is a rational number, then

  1. ln 1 = 0
  2. ln ( a b ) = ln a + ln b
  3. ln ( a b ) = ln a ln b
  4. ln ( a r ) = r ln a

Proof

  1. By definition, ln 1 = 1 1 1 t d t = 0 .
  2. We have
    ln ( a b ) = 1 a b 1 t d t = 1 a 1 t d t + a a b 1 t d t .

    Use u -substitution on the last integral in this expression. Let u = t / a . Then d u = ( 1 / a ) d t . Furthermore, when t = a , u = 1 , and when t = a b , u = b . So we get
    ln ( a b ) = 1 a 1 t d t + a a b 1 t d t = 1 a 1 t d t + 1 a b a t · 1 a d t = 1 a 1 t d t + 1 b 1 u d u = ln a + ln b .
  3. Note that
    d d x ln ( x r ) = r x r 1 x r = r x .

    Furthermore,
    d d x ( r ln x ) = r x .

    Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have
    ln ( x r ) = r ln x + C

    for some constant C . Taking x = 1 , we get
    ln ( 1 r ) = r ln ( 1 ) + C 0 = r ( 0 ) + C C = 0 .

    Thus ln ( x r ) = r ln x and the proof is complete. Note that we can extend this property to irrational values of r later in this section.
    Part iii. follows from parts ii. and iv. and the proof is left to you.

Using properties of logarithms

Use properties of logarithms to simplify the following expression into a single logarithm:

ln 9 2 ln 3 + ln ( 1 3 ) .

We have

ln 9 2 ln 3 + ln ( 1 3 ) = ln ( 3 2 ) 2 ln 3 + ln ( 3 −1 ) = 2 ln 3 2 ln 3 ln 3 = ln 3 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use properties of logarithms to simplify the following expression into a single logarithm:

ln 8 ln 2 ln ( 1 4 ) .

4 ln 2

Got questions? Get instant answers now!

Defining the number e

Now that we have the natural logarithm defined, we can use that function to define the number e .

Definition

The number e is defined to be the real number such that

ln e = 1 .

To put it another way, the area under the curve y = 1 / t between t = 1 and t = e is 1 ( [link] ). The proof that such a number exists and is unique is left to you. ( Hint : Use the Intermediate Value Theorem to prove existence and the fact that ln x is increasing to prove uniqueness.)

This figure is a graph. It is the curve y=1/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1 and to the right at x=e. The area is labeled “area=1”.
The area under the curve from 1 to e is equal to one.

The number e can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series ). Its approximate value is given by

e 2.71828182846 .

The exponential function

We now turn our attention to the function e x . Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by exp x . Then,

Questions & Answers

questions solve y=sin x
Obi Reply
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
follow algebraic method. look under factoring numerator from Khan academy
moe
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask