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

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

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

ln 8 ln 2 ln ( 1 4 ) .

4 ln 2

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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

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
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Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
ayo Reply
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
Ahmad
by using integration product formula
Roha
find derivative f(x)=1/x
Mul Reply
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
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What is the first fundermental theory of Calculus?
ZIMBA Reply
do u mean fundamental theorem ?
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I want simple integral
aparna Reply
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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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