What is the derivative of sech u with respect to x?
Answers
Answer:
\(\displaystyle \frac{dy}{dx} = -u'sech(u)tanh(u)\)
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Step-by-step explanation:
*Note:
This is a known derivative (apply trigonometric differentiation).
Step 1: Define
Identify
\(\displaystyle y = sech(u)\)
Step 2: Differentiate
Hyperbolic Differentiation [Derivative Rule - Chain Rule]: \(\displaystyle y' = -u'sech(u)tanh(u)\)Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Related Questions
For each pair of functions f, g below, find f(g(x)) and g(f(x))
Then, determine whether and are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all in the domain of the composition.
You do not have to indicate the domain.)
Answers
Answer:
See below
Step-by-step explanation:
Part A
\(f(g(x))=f(\frac{x}{3})=3(\frac{x}{3})=x\\g(f(x))=g(3x)=\frac{3x}{3}=x\)
Since BOTH \(f(g(x))=x\) and \(g(f(x))=x\), then \(f\) and \(g\) are inverses of each other
Part B
\(f(g(x))=f(\frac{x+1}{2})=2(\frac{x+1}{2})+1=x+1+1=x+2\\g(f(x))=g(2x+1)=\frac{(2x+1)+1}{2}=\frac{2x+2}{2}=x+1\)
Since BOTH \(f(g(x))\neq x\) and \(g(f(x))\neq x\), then \(f\) and \(g\) are NOT inverses of each other
Jess and Gabby are making posters for the school dance. Jess has made 5 posters. Together, they have made 20 posters. How many posters has Gabby made?
Answers
Let's use the variable "J" to represent the number of posters Jess did, and "G" to represent the number of posters Gabby did.
If they made 20 posters together, we have:
\(J+G=20\)Then, if Jess made 5 posters (that is, J = 5), we can calculate how many posters Gabby did:
\(\begin{gathered} 5+G=20 \\ G=20-5 \\ G=15 \end{gathered}\)So Gabby did 15 posters.
Debbie has 340 dimes in three boxes. she says that there are 4 times as many dimes in the first box as in the second and 3 times as many in the third box as in the first how much money in dollars does she have in each box
Answers
The amount of money in the first box is 8 dollars
The amount of money in the second box is 2 dollars
The amount of money in the third box is 24 dollars
Solving equationsFrom the question, we are to determine how much money there are in each box
From the given information,
"Debbie has 340 dimes in three boxes"
Let x represents the number of dimes in the first box
y represent the number of dimes in the second box
and z represent the number of dimes in the third box
Thus,
x + y + z = 340
Also, from the given information
"she says that there are 4 times as many dimes in the first box as in the second"
x = 4y
and
"3 times as many in the third box as in the first"
z = 3x
Substitute the second and third equations into the first equation
x + y + z = 340
4y + y + 3x = 340
4y + y + 3(4y) = 340
4y + y + 12y = 340
17y = 340
y = 340/17
y = 20
Substitute the value of y into the second equation
x = 4y
x = 4(20)
x = 80
Substitute the value of x into the third equation
z = 3x
z = 3(80)
z = 240
Thus,
There are 80 dimes in the first box
80 dimes = 8 dollars
There are 20 dimes in the second box
20 dimes = 2 dollars
There are 240 dimes in the third box
240 dimes = 24 dollars
Hence,
The amount of money in the first box is 8 dollars
The amount of money in the second box is 2 dollars
The amount of money in the third box is 24 dollars
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11. You have $200 and wish to buy a computer. You find an investment that increases
by 0.6% each month, and you put your $200 into the account. When will the
account enable you to purchase a computer costing $500?
Answers
Answer:
I’m not a hundred percent it’s correct
Step-by-step explanation:
I used the online book to get the answer.
Hope this helps
Using an exponential equation, it is found that the account will enable the person to purchase the computer after 153 months.
An exponential equation of increase is modeled by:
\(A(n) = A(0)(1 + r)^n\)
In which
A(0) is the initial value.r is the rate of increase, as a decimal.In this problem:
$200 put into the account, thus \(A(0) = 200\).Increases 0.6% each month, thus \(r = 0.006\).Then:
\(A(n) = A(0)(1 + r)^n\)
\(A(n) = 200(1 + 0.006)^n\)
\(A(n) = 200(1.006)^n\)
You need $500 to purchase the computer, then:
\(A(n) = 200(1.006)^n\)
\(500 = 200(1.006)^n\)
\((1.006)^n = \frac{500}{200}\)
\((1.006)^n = 2.5\)
\(\log{(1.006)^n} = \log{2.5}\)
\(n\log{1.006} = \log{2.5}\)
\(n = \frac{\log{2.5}}{\log{1.006}}\)
\(n = 153\)
The account will enable you to purchase a computer after 153 months.
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Reduce to simplest form.
-3/2 - 3/8
Answers
Answer:
hope this help you a lot
have a great day
URGENT!!!! Find the y-intercept of the line that passes through the points (-3, 7) and (3, −9).
Answers
Answer:
\(m = \frac{ - 9 - 7}{3 - ( - 3)} = \frac{ - 16}{6} = - \frac{8}{3} \)
\(7 = - \frac{8}{3} ( - 3) + b\)
\(7 = 8 + b\)
\(b = - 1\)
\(y = - \frac{8}{3} x - 1\)
The y-intercept is -1.
If 5, x, y, and 40 are in geometric progression find x and y respectively
Answers
The missing terms in the geometric progression given are:
x = 10 y = 20
What is a Geometric Progression?The nth term of a Geometric progression is expressed by the formula:
an = ar^(n - 1)
Where we have:
an = nth term
a = first term of the geometric progression
n = number of terms
r = common ratio
We are given the geometric progression 5, x, y, and 40, where:
a1 = 5
a4 = 40
Therefore, we have:
40 = 5 * r^(4 - 1)
40 = 5 * r³
40/5 = r³
8 = r³
r = ∛8
r = 2
Since we know a and r, find the 2nd and 3rd which are represented by x and y respectively, by applying an = ar^(n - 1):
a2 = 5 * 2^(2 - 1)
a2 = 5 * 2^1
a2 = 10
x = 10
a3 = 5 * 2^(3 - 1)
a3 = 5 * 2²
a3 = 20
y = 20
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If JK AND LM which statement is true
Answers
Answer:
Perpendicular lines meet at 90 degrees, so JK and LM meeting at 90 degrees is the correct answer.
Step-by-step explanation:
(I know they mean If JK is perpendicular to LM because I saw this question before on brainly.)
you invest 1,000 in an account that pays simple interest of 3% for 10 years. what is the amount of money you'll have at the end of the 10 years?
Answers
Given data:
The given principal is P=1,000.
The given rate of interest is r=3%.
The given time is t=10 years.
The expression for the final amount after 10 years is,
\(A=P+\frac{P\times r\times t}{100}\)Substitute the given values in the above expression.
\(\begin{gathered} A=1,000+\frac{1,000\times3\times10}{100} \\ =1,300 \end{gathered}\)Thus, the final amount after 10 years is 1,300.
Use slopes to determine if the lines 6x + 9y = 2 and 9x + 6y = -3 are perpendicular.
Answers
Explanation:
Make the equation into slope-int form:
6x + 9y = 2
9y = -6x + 2
y = (-6/9)x + 2/9
y = (-2/3)x + 2/9
9x + 6y = -3
6y = -9x - 3
y = (-9/6)x - 3/6
y = (-3/2)x - 1/2
In order for a line to perpendicular the slope needs to be an opposite reciprocal of the other. If our first slope is -2/3 then the opposite reciprocal should be 3/2 but, we found out that our second slope is -3/2 not 3/2 therefore, it will not perpendicular.
Can you explain to me why I got this wrong? And what are the correct answers
Answers
To get the true statements, we will be testing the options one by one in order.
First option
\(\begin{gathered} \cos (25)=0.9063 \\ \cos (65)=0.4226 \\ \therefore\cos (25)\ne\cos (65) \end{gathered}\)Second option
\(\begin{gathered} \cos (25)=0.9063 \\ \sin (65)=0.9063 \\ \therefore\cos (25)=\sin (65) \end{gathered}\)Third option
\(\begin{gathered} \sin (25)=0.4226 \\ \cos (65)=0.4226 \\ \therefore\sin (25)=\cos (65) \end{gathered}\)Fourth option
\(\begin{gathered} \tan (45)=1 \\ \tan (45)=1 \\ \therefore\tan (45)=\tan (45) \end{gathered}\)Fifth option
\(\begin{gathered} \tan (65)=2.1445 \\ \tan (25)=0.4663 \\ \therefore\tan (65)=\tan (25) \end{gathered}\)Hence, the correct options are Options 2, 3, 4.
8. Write a paragraph proof.
Proof Given: In a plane, a is perpendicular to b, b id perpendicular to c, and c || d.
Prove: a || d
Answers
To prove that line segment a is parallel to line segment d, based on the given information, we can utilize the properties of perpendicular and parallel lines.
Given that a is perpendicular to b and b is perpendicular to c, we know that angles formed between a and b, as well as between b and c, are right angles. Let's denote these angles as ∠1 and ∠2, respectively.
Now, since c is parallel to d, we can conclude that the corresponding angles ∠2 and ∠3, formed between c and d, are congruent.Considering the fact that ∠2 is a right angle, it can be inferred that ∠3 is also a right angle.
By transitivity, if ∠1 is a right angle and ∠3 is a right angle, then ∠1 and ∠3 are congruent.Since corresponding angles are congruent, and ∠1 and ∠3 are congruent, we can deduce that line segment a is parallel to line segment d.
Thus, we have successfully proven that a is parallel to d based on the given information and the properties of perpendicular and parallel lines.
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Find the value of x if the figure is a regular polygon.
135
45
1080
1440.
Answers
The value of x is 135. So option(1) is correct.
An angle inside a form is called an interior angle. The polygons have vertices and sides and are closed shapes. Every interior angle of a regular polygon is equal to every other interior angle. For instance, the inner angles of a square are all equal to the right angle, or 90 degrees.
Given figure is a regular polygon
We have to determine the value of x
Since the given polygon has 8 sides
Sum of angle = (2n-4) x 90
= [(2 x 8 – 4)] x 90
= (16-4) x 90
= 12 x 90
= 1080
Therefore each interior angle = 1080/8
= 135
Hence the value of x is 135.
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HELP PLZ
When making bread from scratch, the recipe calls for 3/5 cup of water. If you need to make a smaller portion of the recipe, how much water would you need in order to make only 1/6 th of the recipe?
You would need ___ cups to make the smaller recipe.
Answers
Answer:
3/30 water
Step-by-step explanation:
3/5 water = 1 portion
x = 1/6 portion
x= (1/6 portion) (3/5 water) / 1 portion
x =3/30 water
I will give brainiest to whoever answers correctly !!
Answers
Answer:
14.0
Step-by-step explanation:
14.096 rounded to the nearest tenth
How much is 3 times 4
Answers
Answer:
3×4 = 12!
TWO WAYS TO LOOK AT IT:
(A) 3 groups of 4!
(B) 4 groups of 3 apples!
Amadi is three times as old as Chima. The sum of their ages is 24
Answers
Answer:
Amadi: 20 years old
Chima : 4 years old
PLEASE HELP ME WILL GIVE BRAIN!!
Answers
Student A rewrote the expression correctly. When the factor is multiplied by the quotient, it yields the dividend.
Student B did not rewrite the expression correctly. When the factor is multiplied by the quotient, it does not yield the dividend.
Student C rewrote the expression correctly. When the factor is multiplied by the quotient, it yields the dividend.
The expression when it is rewritten is 4x²(9x + 4x³)
What is a common factor?The factor of a number is a number that perfectly divides a number. A common factor is a factor that two or more numbers share. An example of a common factor of 10 and 20 is 10.
In order to determine if the students rewrote the expressions correctly, use the distributive property to determine if it would yield the initial expression. If it does, the students are correct.
Student A = 4x³(9 + 4x²) = 36x³ + 16\(x^{5}\)
Student B = 3x²(12x + 2x³) = 36x³ + 6\(x^{5}\)
Student C = 2x(18x² + 8\(x^{4}\)) = 36x³ + 16\(x^{5}\)
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Please help quick I need it
Answers
The area of the composite figure is 122.24 units².
How to find the area of a composite figure?The composite figure consist of a rectangle and two semi circles. Therefore, the area of the composite figure is the sum of the area of the individua shapes.
Hence,
area of the composite figure = area of the rectangle + 2(area of semi circle)
Therefore,
area of the composite figure = 9 × 8 + 2(1 / 2 πr²)
area of the composite figure = 72 + πr²
where
r = 8 / 2 = 4 units
Therefore,
area of the composite figure = 72 + 3.14 × 4²
area of the composite figure = 72 + 3.14 × 16
area of the composite figure = 72 + 50.24
area of the composite figure = 122.24 units²
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At Silver Gym, membership is $30 per month, and personal training sessions are $35 each. At Fit Factor, membership is $60 per month, and personal training sessions are $25 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?
Answers
Answer: 5 personal training sessions
Step-by-step explanation:
Given only 1 month, then set the 2 equations equal to each other, given that x=# of personal training sessions:
25+45x = 75 + 35x
-25-35x -25-35x
So, 10x=50; divide both sides by 10, to get:
5, so, 5 personal training sessions per month to make the 2 costs equal to one another.
Sam's net paycheck per month is approximately $3600. Of that amount, he puts $400 into his savings account and uses $200 for entertainment. The rest of the paycheck is used to pay bills. What is the ratio (written as a reduced fraction) of the amount spent on entertainment to the amount spent on bills?
Answers
The ratio of the amount spent on entertainment to the amount spent on bills is 1:15.
What is a ratio?Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
In this case, Sam's net paycheck per month is approximately $3600 and he puts $400 into his savings account and uses $200 for entertainment.
The amount spent on bills will be:
= $3600 - $400 - $200
= $3000
Therefore, the ratio of the amount spent on entertainment to the amount spent on bills is:
= 200/3000
= 1/1
= 1:15
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260 students were surveyed on whether they prefer apple juice or orange juice. A table of relative frequencies is shown. How many more girls prefer apple juice than boys
Answers
Answer:
We need a picture of the screen plz
Step-by-step explanation:
Arrange the steps in order to simplify the expression
Answers
Answer:
Step-by-step explanation:
For step explanation:
1. write the problem
2. distinguishing the neg sign
3. distributing 3
4. moving like terms next to each other through commutative property
5. Combining like terms
6. getting rid of parentheses
HELP ME PLS
Which of these figures has rotational symmetry?
Answers
Answer:
A
Step-by-step explanation:
When you rotate the shape it will be the same as it was before .
If a. = 2.4, b = 3.6, and c = 3.7 what is the value of (a − 1.7 ⋅ b) + a ⋅ c
Please help me :(
Answers
Answer:
5
Step-by-step explanation:
Answer:
5.16 is your answer
Step-by-step explation
fill in your variables with their numbers so
(a - 1.7 * b) + a * c
(2.4 - 1.7 * 3.6) + 2.4 * 3.7
PEMDAS
parentheses first
(2.4 - 1.7 * 3.6)
multiply 1.7 * 3.6 because multiplication is next in PEMDAS
1.7 * 3.6 = 6.12
also multiply your 2.4 * 3.7
2.4 * 3.7 = 8.88
fill in your answers
(2.4 - 6.12) + 8.88
finish your parentheses
2.4 - 6.12 = -3.72
take your answer and move it over
-3.72 + 8.88 =
5.16
P=a+b+c/3 solve for b
Answers
Answer:
3P= 3a+3b+c
3b=3a+c-3P
b= (3a+c-3P)/3
b= (3a)/3+c/3-(3P)/3
b= a+c/3-P
Step-by-step explanation:
let the expression with b be on one side of the equation while you send the other expressions to the other side of the equation.
now, divide through the equation with the coefficient of b which is 3.
simplify the equation by making b the subject of the equation
Find the inverse of the following function. g(x) = 1/x - 2
plz show work
A.) g^-1(x) = - 2/x-1
B.) g^-1(x) = 2/x-1
C.)g^-1(x) = 1/x+ 2
D.) g^-1(x) = 2/x+1
Answers
Answer: C. g^(-1) (x)= 1/(x+ 2)
Step-by-step explanation:
y=1/x-2
x=1/y-2
1/y-2=x
1/y=x+2
1=(x+2)y
1=xy+2y
xy+2y=1
(x+2)y=1
y=1/(x+2)
answer: g^(-1) (x)= 1/(x+ 2)
Mary spends 2 2 3 hours on math homework every week. She also spends 3 1 3 hours on art homework every week. How much time does Mary spend in total on math and art over 4 weeks?
Answers
Answer: This is the answer. please give me the brainliest. Thanks
PLEASE HELP! Mr. Smith makes $20 an hour working full time. He gets about 25% of his income taken out for taxes. He came up with the following monthly budget.
Household:
Rent: $700
Cable: $85
Cell Phone: $175
Electric: $100
Food: $350
Total: 1,410
Automobile
Car: $200
Car Insurance: $100
Gas: $90
Maintenance: $15
Total: $405
How much money does he have left over monthly to put into savings?
Answers
Answer:
$585/month
Step-by-step explanation:
To determine how much money Mr. Smith has left over monthly to put into savings, we need to first calculate his monthly income and his monthly expenses, including taxes.
Mr. Smith's hourly rate is $20, and he works full time, which is typically 40 hours per week. Therefore, his weekly income is:
$20/hour x 40 hours/week = $800/week
To find his monthly income, we multiply his weekly income by the number of weeks in a month:
$800/week x 4 weeks/month = $3,200/month
To find his monthly expenses, we add up the amounts for his household and automobile expenses:
$1,410/month (household) + $405/month (automobile) = $1,815/month
Since Mr. Smith gets about 25% of his income taken out for taxes, we need to calculate the amount of taxes he pays each month:
$3,200/month x 0.25 = $800/month (taxes)
Now we can subtract his monthly expenses and taxes from his monthly income to find out how much money he has left over to put into savings:
$3,200/month - $1,815/month - $800/month = $585/month
Therefore, Mr. Smith has $585/month left over to put into savings after paying his expenses and taxes.
Im diying:(
dont be mean man
Can someone help me with this pleaseeeeee
Answers
The values of x, y , and z in the matrix equation is 3, 4, 0 respectively.
What is the solution of the matrix equation?The solution of the matrix equation is calculated by applying Cramer's rule as shown below;
[ 1 1 -1 ] [ 7 ]
[ 2 3 0 ] [ 18 ]
[ -5 -7 -1 ] [ -43 ]
The determinant of the matrix is calculated as follows;
[ 1 1 -1 ]
[ 2 3 0 ]
[ -5 -7 -1 ]
Δ = 1 (-3 - 0) - 1(-2 - 0 ) - 1(-14 + 15)
Δ = -2
The x determinant of the matrix is calculated as follows;
[ 7 1 -1 ]
[ 18 3 0 ]
[ -43 -7 -1 ]
Δx = 7 (-3 - 0) - 1 (-18 - 0 ) - 1(-126 + 129)
Δx = -6
The y determinant of the matrix is calculated as follows;
[ 1 7 -1 ]
[ 2 18 0 ]
[ -5 -43 -1 ]
Δy = 1 (-18 - 0 ) - 7(-2 - 0 ) -1(-86 + 90)
Δy = -8
The z determinant of the matrix is calculated as follows;
[ 1 1 7 ]
[ 2 3 18 ]
[ -5 -7 -43]
Δz = 1 (-129 + 126) - 1(-86 + 90) + 7(-14 + 15)
Δz = 0
The values of x, y , and z is calculated as;
x = Δx/Δ = -6/-2 = 3
y = Δy/Δ = -8/-2 = 4
z = Δz/Δ = 0/-2 = 0
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