3)y = In (3x-x') 13) y =x (3ln x-1) 4) u= In cos 3t 14) y = x In (1-x) 5) 0 = on tan a 15) y = cot In x 6) w = In Va? -x2 16) y =x In (a*+x)+ka %3D y= logio sin x/a 17) y =x'ln x 8) a = logio (1–4 tan B) 18) y = In1+] %3D
3)y = In (3x-x') 13) y =x (3ln x-1) 4) u= In cos 3t 14) y = x In (1-x) 5) 0 = on tan a 15) y = cot In x 6) w = In Va? -x2 16) y =x In (a*+x)+ka %3D y= logio sin x/a 17) y =x'ln x 8) a = logio (1–4 tan B) 18) y = In1+] %3D
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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SOLVE for items 3, 8 and 13. Find the first derivative and kindly simplify it further. I have provided some information.*see the attached photo.Thank you.
Transcribed Image Text:Exercises: find the first derivative of the following:
1) y = In(7-3x)
11) y = cot In x
2) y = In(2x'+x-1)
12) y = In Inx
v5) y = In (3x-x')
13) y =x' (3ln x -1)
4) u= In cos 3t
14) y = x In (1-x)
5) 0 = on tan a
15) y = cot In x
6) w = In Va -x
16) y =x In (a'+x)+pa Arctan x/a)-2x
Wy= logio sin x/a
17) y =x'In x
18) y = In
.1-t
8) a = logio (14 tan B)
%3D
19) y = logio 4t -1
4t +1
9) y = In 1 +sin x
1-sin x
10) y =x'ln (1+x)
20) y = In'x
%3D
![C. Logarithmic Functions
Logarithm (log)
def n: it states that the logarithm of a number to a certain base, is the
exponent to which the base is raised, to get the said number. It is the inverse of the
exponential functions.
y = log,x if x = a , a>1
Facts about logarithms:
1. negative numbers have no real logarithms
2. numbers between 0 and 1 have negative logarithms
3. numbers greater than 1 have positive logarithms
4. as the number approaches 0, the log becomes negatively infinite
5. the log of 1 is zero
6. as the number becomes infinite, the log is also infinite
Properties of logarithms
a. log, x= log,x - log, y
b. log, xy = log, x+ log, y
c. log, x" =n log, x
d. log, a = x
log X =x
%3D
e. a"
In e =1
Natural base (e) = 2.71828183
Natural logarithm:
natural log with base e are written In u
common log with base 10 are written log u
%3D
2) d. logio u = M (du/dx)
dx
Derivative formula:
1) d. (In u) = du/dx
dx
%3D
Example:
1. y x In x
d. In u du
dx
dx
dy = x'[ 1/x ] + In x (2x)
dx
=x + 2x In x
=x ( 1+2 In x)
2. у%3Dlogio (10-2х)
d. (logio u) = M du/dx
dx
let u = 10-2x
du = -2
u
dy = M[-2]
dx
10-2x
= -2 M.
2[5 -x]
dy =-M.
dx
5-x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefc3f92d-86f5-439e-9dbc-e1d4e3e20b36%2Ff47d5c08-97dd-4102-958f-dbb7f5d778b8%2Fkvpbvf4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:C. Logarithmic Functions
Logarithm (log)
def n: it states that the logarithm of a number to a certain base, is the
exponent to which the base is raised, to get the said number. It is the inverse of the
exponential functions.
y = log,x if x = a , a>1
Facts about logarithms:
1. negative numbers have no real logarithms
2. numbers between 0 and 1 have negative logarithms
3. numbers greater than 1 have positive logarithms
4. as the number approaches 0, the log becomes negatively infinite
5. the log of 1 is zero
6. as the number becomes infinite, the log is also infinite
Properties of logarithms
a. log, x= log,x - log, y
b. log, xy = log, x+ log, y
c. log, x" =n log, x
d. log, a = x
log X =x
%3D
e. a"
In e =1
Natural base (e) = 2.71828183
Natural logarithm:
natural log with base e are written In u
common log with base 10 are written log u
%3D
2) d. logio u = M (du/dx)
dx
Derivative formula:
1) d. (In u) = du/dx
dx
%3D
Example:
1. y x In x
d. In u du
dx
dx
dy = x'[ 1/x ] + In x (2x)
dx
=x + 2x In x
=x ( 1+2 In x)
2. у%3Dlogio (10-2х)
d. (logio u) = M du/dx
dx
let u = 10-2x
du = -2
u
dy = M[-2]
dx
10-2x
= -2 M.
2[5 -x]
dy =-M.
dx
5-x
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