Answers
By def. of the derivative, we have for y = ln(x),
[tex]\displaystyle \frac{dy}{dx} = \lim_{h\to0} \frac{\ln(x+h)-\ln(x)}{h}[/tex]
[tex]\displaystyle \frac{dy}{dx} = \lim_{h\to0} \frac1h \ln\left(\frac{x+h}{x}\right)[/tex]
[tex]\displaystyle \frac{dy}{dx} = \lim_{h\to0} \ln\left(1+\frac hx\right)^{\frac1h}[/tex]
Substitute y = h/x, so that as h approaches 0, so does y. We then rewrite the limit as
[tex]\displaystyle \frac{dy}{dx} = \lim_{y\to0} \ln\left(1+y\right)^{\frac1{xy}}[/tex]
[tex]\displaystyle \frac{dy}{dx} = \frac1x \lim_{y\to0} \ln\left(1+y\right)^{\frac1y}[/tex]
Recall that the constant e is defined by the limit,
[tex]\displaystyle e = \lim_{y\to0} \left(1+y\right)^{\frac1y}[/tex]
Then in our limit, we end up with
[tex]\displaystyle \frac{dy}{dx} = \frac1x \ln(e) = \boxed{\frac1x}[/tex]
In Mathematica, use
D[Log[x], x]
Related Questions
What is the missing constant term in the perfect square that starts with x^2+6xx
2
+6xx, squared, plus, 6, x ?
Answers
To find "c" take the number in front of the "x", which in this case is 6, and divide it by 2, then square it, So, in this case 6/2 = 3 and 32 = 9, so c = 9
(b). Find the values of a and b that make f continuous everywhere.
X^2+6x + 5/x^2-3x-4, X<-1
ax^2- bx +3, -1<=x<=3
2x-a+b, x>=3
Answers
Answer:
a=-13/10
b=-5/2
Step-by-step explanation:
Please help!! I’ll give brainliest
Answers
Answer:
4 and 2/3 from choice A on the promblem
teacher.
1. If the hydrogen ion concentration is a measure of the strength of an acid, how much stronger is
an acid like lemon juice with a pH of 2 than an acid like urine which has a pH of 5.5?
2. A 1998 Pontiac Grand-Am depreciates in value by 18% on average each year. If the car
originally sold for $19995 in 1998, how much would the car be worth in 2012?
3. The newest gossip spreads through Meaghann's high school like wildfire. The number of
students who have heard the news triples every two minutes, thanks mostly to texting. If the
gossip started with just one person, how long would it take before the entire school of 1200
students hears the news?
Answers
The three questions are illustrations of arithmetic operations and geometric progressions
(1) Acid concentration
The given parameters are:
pH of acid lemon juice = 2
pH of acid urine = 5.5
The difference between their pH is calculated as follows:
Difference = 5.5 - 2
Difference = 3.5
Multiply the difference by 10, to calculate the acidity
Acidity = 3.5 * 10
Acidity = 35
Hence, the acid like lemon juice is 35 times stronger than urine
(2) Depreciation
The depreciation is given as:
r = 18%
In 1998, the car's worth is $19995.
This is represented as:
a = 19995
2012 is 14 years from 1998.
This is represented as:
n = 14
So, the worth of the car in 2014 is calculated using:
[tex]\mathbf{Worth =a(1 -r)^n}[/tex]
So, we have:
[tex]\mathbf{Worth =19995 \times (1 -18\%)^{14}}[/tex]
Express percentage as decimal
[tex]\mathbf{Worth =19995 \times (1 -0.18)^{14}}[/tex]
[tex]\mathbf{Worth =19995 \times 0.82^{14}}[/tex]
Evaluate
[tex]\mathbf{Worth =1243}[/tex]
Hence, the worth of the car in 2012 is $1243
(3) Spread of Gossip
The spread of the gossip is an illustration of geometric progression, where
a = 1 ---- the first term
r = 3 --- the common ratio
L =1200 --- the last term
So, we calculate the number of terms from 1 to 1200 using the following nth terms of geometric progression
[tex]\mathbf{L = ar^{n-1}}[/tex]
This gives
[tex]\mathbf{1200 = 1 \times3^{n-1}}[/tex]
[tex]\mathbf{1200 = 3^{n-1}}[/tex]
Take logarithm of both sides
[tex]\mathbf{log(1200) = log(3^{n-1})}[/tex]
Apply law of logarithm
[tex]\mathbf{log(1200) = (n-1)log(3)}[/tex]
Divide both sides by log(3)
[tex]\mathbf{n - 1 = log(1200) \div log(3)}[/tex]
[tex]\mathbf{n - 1 =6.45}[/tex]
Add 1 to both sides
[tex]\mathbf{n =7.45}[/tex]
Since the gossip spreads every two minutes.
The total time is:
[tex]\mathbf{Total =2minutes \times 7.45}[/tex]
[tex]\mathbf{Total =14.9\ minutes}[/tex]
Hence, it will take 14.9 minutes for the gossip to spread to 1200 students
Read more about geometric progressions at:
https://brainly.com/question/14320920
y=-1/2x+4 y=6
solve systems of equations using substitution method
BEST ANSWER GETS BRAINLY
Answers
You can plug that in first equation
6 = -1/2x + 4
Subtract 4
2 = -1/2x
Multiply by 2
4 = -1x, x = -4
Solution: x = -4, y = 6
Graph this please any math expert please thank you
Answers
Answer:
we cant see the picture but if i could i would have solved it
Step-by-step explanation:
we cant see the picture n
A committee including 2 boys and 2 girls is to be formed from a group of 6 boys and 5 girls. How many different committee can be formed from the group
Answers
Step-by-step explanation:
well, we need to multiply the possibilities of forming a group of 2 boys out of 6 with the possibilities to firm a group of 2 girls out of 5.
the first is 6 over 2
6! / (2! × (6-2)!) = 6×5/2 = 3×5 = 15
the second one is 5 over 2
5! / (2! × (5-2)!) = 5×4/2 = 5×2 = 10
so, these 15 possibilities combined with the 10 possibilities makes
15×10 = 150
possibilities to form this committee.
Translate the following verbal statement into an algebraic equation and then solve:
Mariah bought a bookcase on sale for $340, which was two-fifths of the original price. What was the
original price of the bookcase?
Answers
Answer:
x times 0.4 = 340
x = 850
Step-by-step explanation:
340 / 0.4 = 850
need help ASAP it due today
Answers
Answer:
3/4
Step-by-step explanation:
25
A man earns N 5,000.00 for 10
days work. How should he earn in
14 days, if working at the same
rate
Answers
Answer:
N 7,000.00
Step-by-step explanation:
He earns N 5,000.00 for 10 days of work so divide the 5,000.00 by 10
5,000.00/10= 500.00 ; this is the amount of money he makes in 1 day
Then multiply this amount make in 1 day by 14 to find how much the man makes in 14 days.
500.00*14=7,000.00
Therefore, the man makes N 7,000.00 in 14 days.
4. Provide an appropriate response. Round to the nearest hundredth.
(1 point)
In a certain town, 40% of adults have a college degree. If four adults from the town are
selected at random, the probabilities that 0, 1, 2, 3, or 4 of them will have college degrees are
provided in the accompanying table. Find the standard deviation for the probability
distribution
Х P(x)
0 0.1296
1 0.3456
2 0.3456
3 0.1536
4 0.0256
Answers
According to the discrete distribution given, the standard deviation is of: 0.98.
The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability, hence:
[tex]E(X) = 0.1296(0) + 0.3456(1) + 0.3456(2) + 0.1536(3) + 0.0256(4) = 1.6[/tex]
The standard deviation is the square root of the sum of the multiplication of each probability and the difference squared between the observation and the mean, hence:
[tex]\sqrt{V(X)} = \sqrt{0.1296(0-1.6)^2 + \cdots + 0.1536(3-1.6)^2 + 0.0256(4-1.6)^2} = 0.98[/tex]
The standard deviation is of 0.98.
A similar problem is given at https://brainly.com/question/25650097
change 480 cm to metres
Answers
I really need help and I would appreciate an explanation but it’s not needed thank you
Answers
125 feet in 5seconds
Answers
Answer:
25 feet in one second
Step-by-step explanation:
Since it takes 5 seconds to travel 125 feet, you would do 125 ÷ 5 to get the distance for one second, which is 25 feet.
Hope it helped!
8 singles, 6 fives, 3 twenties, and 3 hundred dollar bills are all placed in a hat. If a
player is to reach into the hat and randomly choose one bill, what is the fair price to
play this game?
Answers
The fair price to play the game as described is therefore; $19.9.
The fair price to play the game would be the average dollar amount bill.
This can be evaluated as follows;
8 singles, 6 fives, 3 twenties, and 3 hundred dollar bills can be written as;
(8×1) + (6×5) + (3×20) + (3×100)Total amount in the hat = $398The total number of bills = 8+6+3+3 = 20The fair price = $398/20The fair price to play the game as described is therefore; $19.9.
Read more;
https://brainly.com/question/5899088
How would I find the domain and range of this graph
Answers
the domain is the min/max amount of x values
the range is the min/max amount of y values
A soft drink machine outputs a mean of 28 ounces per cup. The machine's output is normally distributed with a standard deviation of 3 ounces. What is the probability of putting less than 33 ounces in a cup? Round your answer to four decimal places.
Answers
Answer:
0.7745
Step-by-step explanation:
19. Select numbers to create a different
expression that is equal to 7 x 9.
2
3
4
6
8
9
7x9=(x 9)+(x)
Answers
Step-by-step explanation:
7 × 9 = (? × 9) + (? × ?)
7 × 9 = (3 × 9) + (4 × 9) or (4 × 9) + (3 × 9)
The required numbers are 3, 4, and 9 to create the expression that is equal to 7 × 9.
What are Arithmetic operations?Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions.
* Multiplication operation: Multiplies values on either side of the operator
For example 4*2 = 8
What is the number system?A number system is defined as a way to represent numbers on the number line using a set of symbols and approaches. These symbols, which are known as digits, are numbered 0 through 9. Based on the basic value of its digits, different types of number systems exist.
We have been given numbers are 2, 3, 4, 6, 8, and 9.
To determine numbers create a different expression that is equal to 7 × 9.
Here substitute the numbers 3, 4, and 9, and we get
⇒ 7 × 9 = (3 × 9) + (4 × 9) = 63
⇒ 7 × 9 = (4 × 9) + (3 × 9) = 63
Therefore, the required numbers are 3, 4, and 9 to create the expression that is equal to 7 × 9.
Learn more about the number system here:
https://brainly.com/question/21751836
#SPJ2
What is the prime factorization of 990?
2 x 3 x 52 x 11
2 x 3 x 5x 11
2 x 32 x 5 x 11
22 x 3 x 5x 11
Answers
Answer:
2 x 3² x 5 x 11
Step-by-step explanation:
990 ÷ 2 = 495 so 2 is one factor
495 ÷ 2 = 247.5 - There is a remainder. So we can't divide by 2 anymore. Use next prime number
495 ÷ 3 = 165 - No remainder. So 3 is one factor
165 ÷ 3 = 55 - No remainder. So 3 is one factor
55 ÷ 3 = 18.3333 - There is a remainder. We can't divide by 3. Use next prime number
55 ÷ 5 = 11 - No remainder. So 5 is one factor
11 ÷ 5 = 2.2 - There is a remainder. We can't divide by 5. Use next prime number
11 ÷ 7 = 1.5714 - This has a remainder. So 7 is not a factor.
11 ÷ 11 = 1 - No remainder. SO 11 is one factor
Now put it together: 2 x 3² x 5 x 11
Factor trees also help. Have a good day or night
i neeeed help plz help me
Explain how the magnetic field line data supports or refutes the claim that the launcher and spacecraft magnets were misaligned in the Tuesday launch. Refer to the words and phrases below to help you write your response.
Answers
Answer:
the units were only misalignes by a few cm
Landon drove 40 miles in 1 1/4 hours. If he drove at a constant rate, how far did he travel in one hour?
Answers
9514 1404 393
Answer:
32 miles
Step-by-step explanation:
To find miles per hour, divide miles by hours.
40 mi/(1.25 h) = 32 mi/h
At that rate, Landon traveled 32 miles in 1 hour.
Sara made a withdrawal of 80$. Tom deposited 47.50.. which transactions involved more money?
Answers
Answer:
I don’t know bums
Step-by-step explanation:
y’all dum
Jackson works at a company that makes widgets. If the machine he runs can make 36 widgets in 3
hours, how many hours does it take his machine to produce 66 widgets?
Answers
Answer:
Jackson needs 30 more widgets 2 hours
Step-by-step explanation:
Pls and thank you, God bless you
Answers
Answer:
100N to the right
Step-by-step explanation:
Hope that this is helpful.
Have a great day.
I need help I don’t understand this question
Answers
Answer:
-1
Step-by-step explanation:
0 - (-1) + (-2) = 1 - 2 = -1
simplify this algebraic expression completely 5y - 3(y+2)
Answers
Answer: 2y - 6
Step-by-step explanation:
This problem utilizes the Distributive Property of Multiplication. It states that: a(x + y) = ax + ay.
Thus, utilizing this property (distributing -3), change the equation into:
5y - 3y - 6
Then, simply combine like terms to get:
2y - 6
Hope it helps :) and let me know if you want me to elaborate.
help me in zearn please
Answers
Answer:
Diameter: 25.6
Circumference: 80.38
Note: Enter your answer and
Use matrices to solve the system.
2x + 3y = 4
5x + 8y = 11
Need help solving this
Answers
9514 1404 393
Answer:
(x, y) = (-1, 2)
Step-by-step explanation:
The augmented matrix for this system of equations is ...
[tex]\left[\begin{array}{cc|c}2&3&4\\5&8&11\end{array}\right][/tex]
Your graphing calculator or web solver will tell you the reduced row-echelon form of this matrix is ...
[tex]\left[\begin{array}{cc|c}1&0&-1\\0&1&2\end{array}\right][/tex]
The solution is (x, y) = (-1, 2).
_____
The determinant of the coefficient matrix is ...
(2)(8) -(5)(3) = 1
The inverse of the coefficient matrix is the transpose of the cofactor matrix, divided by the determinant. For a 2×2 matrix, that is the original matrix with the diagonal terms swapped, and the off-diagonal terms negated, then divided by the determinant. Since the determinant is 1, the inverse of the coefficient matrix is ...
[tex]\left[\begin{array}{cc}8&-3\\-5&2\end{array}\right][/tex]
Multiplying this by the constant vector gives the solution vector.
x = (8)(4) +(-3)(11) = 32 -33 = -1
y = (-5)(4) +(2)(11) = -20 +22 = 2
(x, y) = (-1, 2)
_____
Additional comment
Many modern calculators have so many functions that there are likely many you will never use. It is a good idea to learn to use the functions available for solving the kinds of problems you run across. Recent model calculators make it easy to enter matrices and perform various functions on them.
For solving systems of linear equations, the RREF (reduced row-echelon form) function is pretty useful. The bottom row of the result will tell you the kinds of solutions; the right column(s) will tell you what they are.
YALL PLS HELP IM GOVE BRAINLIST PLSSS
Answers
Answer:
If the wheels of diameter of 20 inches rotate 37 times then she will travel 2325.71 inches or 59 m.
Step-by-step explanation:
pls mark as brainliest need to rank up
:: The radius, r=2 = 10 inches
We know that 1 rotation of the wheel is equal to
the circumference of the wheel.
Circumference of the wheel =211 =2* #7*10
= 62.85 inches
Here, it is given that Jada's bike's wheels rotate
for 37 times.
Thus,
The distance travelled by Jada if the wheel
rotates for 37 times will be,
= 37* Circumference
= 37 * 62.85
= 2325.45 inches or 59 m
Margo can purchase tile at a store for $0.99 per tile and rent a tile saw for $18. At another store she can
borrow the tile saw for free if she buys tiles there for $1.29 per tile. How many tiles must she buy for the
cost to be the same at both stores?
Answers
Answer:
.79x+24=1.29
took way long then it should my fault but 1.29-.79=.5
x=18/.5
x=36 sorry for taking so long and this the answer i just answered it in a weird way sorry
Step-by-step explanation:
