Answers
Answer:
[tex] \displaystyle{ y' = \dfrac{2}{1 - {x}^{2} }}[/tex]
Step-by-step explanation:
Apply natural logarithm chain rule:
[tex] \displaystyle{y = \ln(u) \to y' = \dfrac{u'}{u}}[/tex]
Where u is a function. The ' symbol indicates that the function to be derived. From this formula, we will have:
[tex] \displaystyle{y' = \dfrac{ \left (\dfrac{1 + x}{1 - x} \right)'}{ \dfrac{1 + x}{1 - x}}}[/tex]
Now we have to derive quotient functions, we can use quotient rule for this:
[tex] \displaystyle{y = \dfrac{u}{v} \to y' = \dfrac{u'v - uv'}{ {v}^{2} }}[/tex]
Where u and v are functions. Therefore, from numerator:
[tex] \displaystyle{ \left( \dfrac{1 + x}{1 - x} \right)' = \dfrac{(1 + x)'(1 - x) - (1 + x)(1 - x)'}{ {(1 - x)}^{2} }}[/tex]
Derive using power rule and constant rule where:
[tex] \displaystyle{y = {x}^{n} \to y' = n {x}^{n - 1} } \\ \\ \displaystyle{y = c \to y' = 0 \: \: \: \: ( c \in \mathbb{R})}[/tex]
Thus:
[tex] \displaystyle{ \left( \dfrac{1 + x}{1 - x} \right)' = \dfrac{1(1 - x) - (1 + x)( - 1)}{ {(1 - x)}^{2} }} \\ \\ \displaystyle{ \left( \dfrac{1 + x}{1 - x} \right)' = \dfrac{1 - x- ( - 1 - x)}{ {(1 - x)}^{2} }} \\ \\ \displaystyle{ \left( \dfrac{1 + x}{1 - x} \right)' = \dfrac{1 - x + 1 + x}{ {(1 - x)}^{2} }} \\ \\ \displaystyle{ \left( \dfrac{1 + x}{1 - x} \right)' = \dfrac{2}{ {(1 - x)}^{2} }}[/tex]
Therefore, we will now have:
[tex] \displaystyle{y' = \dfrac{ \dfrac{2}{ {(1 - x)}^{2} }}{ \dfrac{1 + x}{1 - x}}}[/tex]
Simplify expressions:
[tex] \displaystyle{ \dfrac{ \dfrac{2}{ {(1 - x)}^{2} }}{ \dfrac{1 + x}{1 - x}} = \dfrac{2}{ {(1 - x)}^{2}} \times \dfrac{1 - x}{1 + x} } \\ \\ \displaystyle{ \dfrac{ \dfrac{2}{ {(1 - x)}^{2} }}{ \dfrac{1 + x}{1 - x}} = \dfrac{2}{ 1 - x} \times \dfrac{1 }{1 + x} } \\ \\ \displaystyle{ \dfrac{ \dfrac{2}{ {(1 - x)}^{2} }}{ \dfrac{1 + x}{1 - x}} = \dfrac{2}{ (1 - x)(1 + x)}}[/tex]
Therefore, the derived function is:
[tex] \displaystyle{ y' = \dfrac{2}{1 - {x}^{2} }}[/tex]
Related Questions
A ride at a carnival moves on a circular path with diameter 80ft. What is the approximate distance of one complete loop?
Answers
Answer:
80π feet
Step-by-step explanation:
The question is asking us for the circumference of a circle with diameter 80 ft.
The formula for the circumference of a circle is πd. Therefore, the circumference is π * 80 = 80π ft!
if this helps you, it would help me a lot if you could mark this as brainliest :)
Answer:
≈ 251 ft
Step-by-step explanation:
the distance covered in one complete cycle is the circumference of the circle.
the circumference (C) is calculated as
C = πd ( d is the diameter ) , then
C = π × 80 ≈ 251 ft ( to the nearest whole number )
Combine the like terms to create an equivalent expression:
−3n−7+(−6n)+1
Answers
Answer:
-9n - 6
Step-by-step explanation:
1. combine the n's
-3n - 6n = -9n
2. combine the constants
-7 + 1 = -6
Brooklyns mother gave her an allowance at the beginning of a certain summer break. Brooklyn spent an equal amount of money each week from the allowance. The line graph shows her allowance over 4 weeks. How much did brooklyn spend on her allowance each week
Pls help!! Iv'e been struggling for a while!
Answers
Answer: wait how much was her allowance
Step-by-step explanation:
two airplanes are flying in the air at the same height: airplane a is flying east at $$ mi/h and airplane b is flying north at $$ mi/h. if they are both heading to the same airport, located $$ miles east of airplane a and $$ miles north of airplane b, at what rate is the distance between the airplanes changing?
Answers
The distance between the airplanes changing is -390.
Calculation:The miles from the airport to Airplane A can be calculated as follows: a = 30 -250t, where t is in hours.
Similar to airplane A, airplane B's distance from the airport can be expressed as...b = 40 -300t
Using the Pythagorean theorem, one can calculate the separation (d) between the aircraft:
d^2 = a^2 + b^2
Depending on the passage of time, we have...2dd' = 2aa' + 2bb', and d' = (aa' + bb')/d.
The values of its variables at t=0 must be determined in order to determine the numerical value of thisa = 30 -2500 = 30 a' = -250 b = 40 -3000 = 40 b' = -300
d = √(a²+b²) = √(900+1600) = 50
Then, d' = (30(-250) +40(-300))/50 = -19500/50 = -390
At a speed of 390 miles per hour, the space between the aircraft is closing.
How far apart are the equation's two planes?The distance between the two planes is going to be the square root of six, therefore if we solve for d, multiply both sides of this equation by the square root of six, we obtain six is equal to negative d, or d is equal to negative six.
How widely apart are commercial airplanes?For a commercial airliner, separation will typically be at least 3 miles laterally or 1,000 feet vertically (as stated in the question). The 5 mile rule is applied laterally in the enroute environment, at greater operating speeds over 10,000 feet, depending on the kind of radar and distance from the antennae.
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the 95% two-sided confidence interval for the mean is (-0.1,0.2). one of your friends claims that the mean could be 0. is your friend's claim reasonable? if yes, select true. if no, select fals
Answers
The 95% two-sided confidence interval for the mean is (-0.1,0.2). one of your friends claims that the mean could be 0. is your friend's claim reasonable.
It is TRUE statement.
Now, According to the question:
Let's know:
What is meant by 95% confidence interval?
Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value (μ).
OR
A confidence interval, in statistics, refers to the probability that a population parameter will fall between a set of values for a certain proportion of times. Analysts often use confidence intervals than contain either 95% or 99% of expected observations.
Hence, The 95% two-sided confidence interval for the mean is (-0.1,0.2). one of your friends claims that the mean could be 0. is your friend's claim reasonable.
It is TRUE statement.
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Which one is not criteria of congruence * ASA SAS AAS SSA?
Answers
Answer:
Step-by-step explanation:
well this makes no sese is there any pics
Eh I don't know ???
Picture should be shown...
Help with question 12
Answers
Answer:
The lowest common denominator is 8
Step-by-step explanation:
To find the lowest common denominator, we look at the multiples
Multiples of 4: 4,8,12,16,20
Multiples of 8: 8,16,24
The 1st common multiple is 8
The lowest common denominator is 8
h equation could be represented by the number line A. 1+ (-7) = -6 B. -7 + 6 = -1 C. -6 + (-1) = -7 D. -6 + 7 = 1
Answers
The equation that could be represented by the number line is 1+ (-7) = -6. Option A. is the answer
How to determine which equation could be represented by the number line?A number line is a line on which numbers are marked at intervals, which is used for illustrating simple numerical operations
Looking at the number line in the image, the arrow from right to left (backward) by 7 steps (-7).
Thus, the equation can be written as 1 + (-7) = -6
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the temperature inside the lab refrigerator is no more than 35 . use t to represent the temperature (in ) of the refrigerator.
Answers
The given situation can be represented in inequality as t ≤ 45 °F.
What is inequality?An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way.
The majority of the time, size comparisons between two numbers on the number line are made.
So, we are instructed to use t to denote the refrigerator's temperature (in°F).
Additionally, we are informed that the lab refrigerator's maximum temperature is 45 °F.
It must be either lower than or equal to 45 °F for the temperature inside the refrigerator to be no higher than that.
As a result, t must be lower than or equal to 45 °F.
Therefore, it can be expressed as inequality as follows:
t ≤ 45 °F
Therefore, the given situation can be represented in inequality as t ≤ 45 °F.
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Complete question:
The temperature inside the lab refrigerator is at most 35 Fahrenheit used to represent the temperature in Fahrenheit of the refrigerator in inequality.
if triangle pqrs is similar to triangle wxyz find the value of x?
Answers
Answer:
x=60
Step-by-step explanation:
these shapes seem to be flipped
side WZ seems to be similar to PS
and Side ZY seems to be similar to RS
ewe can make ratios on the side lengths
40/75=32/x
cross multiply
40x=75(32)
40x=2400
/40. /40
x= 60
hopes this helps please mark brainliest
What’s your least fav food first one gets more points
Answers
My least favorite food would defiently be soup.
Answer:
Step-by-step explanation:
Brussels sprouts
Which of the following is linear equation in two variables?
1. x/3 + 5/y = 6
2. 2x^2 - 3y = 8 - 3y
3. x + 2y = 5 - 3y
4. 3x^2 + y = 0
Answers
The other two answer choices have an x squared term which makes those lines quadratics and not linear.
A linear function is one that can be written with a constant multiplied by a variable as shown by
y = mx + b
The x cannot be squared.
solve for x.
2.4x - 9.1x + 12.5x + = -39.44
Answers
The equation can be written as: 2.4x - 9.1x + 12.5x = -39.44. x = 9.44.
What is distributive property?The distributive property is a mathematical rule that states that multiplying a number by a group of numbers in a set can be done by multiplying the number by each individual number in the set. It is sometimes referred to as the distributive law of multiplication. This property is used to simplify complex expressions and equations into simpler ones. For example, the distributive property allows us to take the expression (x+y) * 5 and rewrite it as x * 5 + y * 5. This makes it easier to work with and solve. The distributive property can be used with addition and multiplication and is one of the most important and useful tools in algebra.
To solve for x, we need to use the distributive property to combine the like terms on the left side of the equation:
2.4x - 9.1x + 12.5x = (2.4 - 9.1 + 12.5)x
= -4.2x
We can now divide both sides of the equation by -4.2 to isolate x:
-4.2x / -4.2 = -39.44 / -4.2
x = 9.44
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How do you evaluate the definite integral ∫2^x dx from [−1,1]?
Answers
The solution for definite integral ∫2^x dx from [−1,1] is
3/ 2 ln2
Definite Integral:
A definite integral is defined as the exact limit and summation examined in the previous section to find the net area between the function and the x axis. Also note that the definite integral notation is very similar to the indefinite integral notation. The reason for this will become clear in due course.
There are a few terms that should be left out along the way. The number "a" below the integral sign is called the lower integral limit, and the number "b" above the integral sign is called the upper integral limit. Even if the intervals a and b are specified, the lower bound is not necessarily less than the upper bound. Collectively, a and b are often referred to as the interval of integration.
Given a function :
f(x) that is continuous function on the interval [a , b]
we divide the interval into n subintervals of equal breadth, Δx, and from each interval choose a point, x i. Then the definite integral of f(x) from a to b is
[tex]\int\limits^b_a f(x) \, dx = \lim_{n \to \infty} f(xi)[/tex] Δx.
According to the question:
let's start by differentiating 2ˣ.
y = 2ˣ
⇒ ln y = x ln 2
⇒ 1/y dy/dx = ln2
⇒ d /dx = y ln 2 = 2ˣ ln2
So remembering integration is the reverse of differentiating.
y = 2ˣ
⇒ dy/dx = 2ˣ ln 2
⇒ ∫2ˣ dx = 2ˣ / ln 2 + C
2ˣ > 0, ∀ x ∈ R, there is no negative area so we can
Insert the limits and evaluate immediately.
= 1/ ln2 [2x][tex]\left \{ {{y= -1} \atop {x=1}} \right.[/tex]
= 1/ln 2 (2¹ - 2⁻¹)
= 3/ 2ln 2
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the systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmhg and a standard deviation of 12 mmhg. approximately what percentage of 18-year-old women have a systolic blood pressure between 108 mmhg and 132 mmhg?
Answers
Percentage of 18-year-old women have a systolic blood pressure between 108 mm hg and 132 mm hg is 68.27 %
z - score :
mean = 120
standard deviation = 12
you want to find the area under the distribution curve between the scores of 108 and 132 .
z - score low :
low z-score = ( 108 - 120 ) / 12 = -12 / 12 = -1.
z - score high :
high z-score = (132 - 120) / 12 = +12 / 12 = +1.
the + sign is normally silent (not shown) but i show it here to emphasize that you are going from a low z-score of -1 to a high z-score of +1.
so here probability = 0.6827
= 0.6827 * 100 %
= 68.27 %
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two less than the product of a number and three
Evaluate when f = 4.5.
A 2-column table with 5 rows. Column 1 is labeled Key Words with entries two, less than, product, a number, three. Column 2 is labeled Replace with entries 2, minus, times, f, 3.
Write and evaluate the expression shown on the left. Then, check all that apply.
Write the expression as 2 – 3f.
Write the expression as 3f – 2.
Write the expression as 2f – 3.
The value when f = 4.5 is 6.75.
The value when f = 4.5 is 6.
The value when f = 4.5 is 11.5.
Answers
The expression as 3f – 2.
The value when f = 4.5 is 11.5.
How to illustrate the equation?An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario
Two less than the product of a number and three will be:
(3 × f) - 2
= 3f - 2
When f = 4.5
(3 × 4.5) - 2
= 11.5
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how many megawatt hours (mwh) per year of savings did the students identify for computers at the penn state mueller laboratory building? how many megawatt hours (mwh) per year of savings did the students identify for computers at the penn state mueller laboratory building? 12 mwh 10 mwh 120 mwh 30 mwh
Answers
There are 30-megawatt hours (MWh) per year of savings the students had identified for computers at the Penn State Mueller Laboratory Building.
"Information available from the question"
How many megawatt hours (mwh) per year of savings did the students identify for computers at the Penn state Mueller laboratory building?
Now, According to the question:
There are 30-megawatt hours (MWh) per year of savings the students had identified for computers at the Penn State Mueller Laboratory Building.
At Pennsylvania State University, an undergraduate class studied an environmental evaluation in one of their laboratory buildings in biology. They want to answer the question, "How is this building like an ecosystem?" Their evaluation result was a report that contains 52-page noting ways to lessen the Mueller Laboratory Building’s environmental footprint in of water, energy communications, maintenance, renovation, and food usage. The students establish techniques to save an almost $45,500 yearly in the energy cost alone for a building with 123 scientists and backing staff.
Hence, There are 30-megawatt hours (MWh) per year of savings the students had identified for computers at the Penn State Mueller Laboratory Building.
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what is the percent of change 8 to 2?
Answers
The percent of change 8 to 2
How do you convert 25 degrees into radians?
Answers
Answer:
25° × π/180 is = 0.4363rad
Step-by-step explanation:
in equilateral triangle abc the length of side a
Answers
In equilateral triangle ABC, the length of side A is equal to the side B and side C.
What is triangle?A triangle is a three-edged polygon with three vertices. It is a fundamental form in geometry. Triangle ABC denotes a triangle with vertices A, B, and C. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane. Triangles are three-sided polygons with three vertices. The angles of the triangle are formed by connecting the three sides end to end at a point. The sum of the triangle's three angles equals 180 degrees.
Here,
The length of side A of an equilateral triangle ABC is equal to the lengths of sides B and C.
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Tyler says that since triangle ACD is similar to triangle ABC, the length of CB is 11.96. Noah says that since ABC is a right triangle, we can use the Pythagorean Theorem. So the length of CB is 12 exactly. Do you agree with either of them?
Answers
I agree with Noah because she used Pythagorean theorem.
What is a Pythagorean theorem?Pythagoras, a Greek philosopher who was born in 570 BC, is honored by having his theorem named after him. The theorem has likely been demonstrated the most times of any mathematical theorem using a variety of techniques.
In analytic geometry, when Euclidean space is represented by a Cartesian coordinate system, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be applied to higher-dimensional spaces, non-Euclidean spaces, objects other than right triangles, and even things that are not triangles at all but n-dimensional solids.
Given that,
Taylor says that triangle ACD is similar to triangle ABC
And length of CB is 11.96
Abd also given that,
Noah says that since ABC is a triangle.
Length of CB is 12
I agree with Noah because she used Pythagorean theorem,
BC²+AC²=AB²
12²+5²=13²
144+25=169
169=169
So, I agree with Noah.
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Which of the following is/are not a quadratic equation?
A. x^2–8x+5=0
B. 3x^2–4=0
C. 2x=x^2
D. x+5=3
Answers
(C) 2x=x^2 is not a quadratic equation and the rest of the equation is a quadratic equation.
What is a quadratic equation?Any algebraic statement that can be represented in a standard form, where x stands for an unknown number, a, b, and c stand for known values, and where a 0 is true, is said to be a quadratic equation.
Factoring, using square roots, completing the square, and using the quadratic formula are the four approaches of solve a quadratic problem.
A quadratic equation's form is: ax²+bx+c=0
(A) x^2–8x+5=0
In the form ax2+bx+c=0.
(B) 3x^2–4=0
In the form ax2+bx+c=0.
(C) 2x=x^2
Not in ax2+bx+c=0 form.
(D) x+5=3
In the form ax2+bx+c=0.
Therefore, (C) 2x=x^2 is not a quadratic equation and the rest of the equation is a quadratic equation.
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Which of the following are quadratic equations? (i) 16x2−3=(2x+5)(5x−3) (ii) (x+2)3=x3−4 (iii) x(x+1)+8=(x+2)(x−2)
Answers
Answer:
Step-by-step explanation:
Rearranging:
(i) 16x2−3=(2x+5)(5x−3)
16x^2 - 3 = 10x^2 + 19x -15
6x^2 - 19x + 12 = 9
THIS IS QUADRATIC.
(ii) (x+2)3=x3−4
(x + 2)(x + 2)^2 = x^3 - 4
(x + 2)(x^2 + 4x + 4) = x^3 - 4
x^3 + 4x^2 + 4x + 2x^2 + 8x + 8 = x^3 - 4
6x^2 + 12x + 4 = 0.
THIS IS QUADRATIC.
(iii) x(x+1)+8=(x+2)(x−2)
x^2 + x + 8 = x^2 - 4
x + 12 = 0
THIS IS NOT QUADRATIC.
Answer:
3
Step-by-step explanation:
because it consists two side separated by equal sign and has letters in it
Please help, giving thanks and brainliest:) (Please give an explanation if possible and please dont give a fake answer)
Answers
In the figure given the value of x is calculated to be
x = 128x = 2x + 72 = 96 degreesHow to find the value of x in the figureThe value of x is solved from using the vertical angle theorem.
According to this theorem, when two straight lines cross, they create two sets of linear pairs with equal angles.
The adjacent angles created by the junction of these two lines are likewise said to be supplementary, or equivalent to 180 degrees.
From this principle,
8x = 2x + 72
8x - 2x = 72
6x = 72
x = 16
The angles marked by 8x is equal to
= 8 * 12
= 96 degrees
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a cylinder is inscribed in a right circular cone of height 5.5 feet and radius (at the base) equal to 3.5 feet. what are the dimensions of such a cylinder which has maximum volume?
Answers
Answer:
Step-by-step explanation:
If r and h radius and of inscribed cylinder then from similyarity of triangles(2- h)/2 = 2r/15, from here h = 2 - 4r/15.
Volume V = πr2h = πr2(2 - 4r/15)= π(2r2 - 4r3/15). Let find derivative and set it to 0:
V' = π(4r - 12r2/15) = 0; we get equation 4r - 4r2/5 = 0, r = 0 and r = 5.
To check if we get maximum we find V'' = π(4 - 8r/5) and at r = 5 V'' = π(4 -8) = -4π < 0 max
Answer: demantions of inscribed cylinder r = 5 and h = 2 - 4·5/15 = 2 - 4/3 = 2/3
ASAP! If a certain cannon is fired from a height of 8.1 meters above the ground, at a certain angle, the height of the cannonball above the ground, h, in meters, at time, t, in seconds, is found by the function h(t) = -4.9t^2. + 27.5t + 8.1. Find the time it takes for the cannonball to strike the ground.
The cannonball will strike the ground after about [] seconds.
(Type an integer or a decimal. Round to the nearest hundredth as needed.)
Answers
The time it takes for the cannonball to strike the ground if a certain cannon is fired from a height of 8.1 meters above the ground, at a certain angle, the height of the cannonball above the ground, h, in meters, at the time, t, in seconds, is found by the function h(t) =-4.9 t² + 27.5t + 8.1 is 5.612 seconds.
What is velocity?A vector measure of movement's speed and direction is called velocity. Simply described, velocity is the rate of movement in a specific direction.
Given:
The height of the cannon = 8.1 m
The height equation, h = -4.9 t² + 27.5t + 8.1
Calculate the time by putting the value of height in the equation as shown below,
h = (-4.9 t² + 27.5t + 8.1)
8.1 = (-4.9 t² + 27.5t + 8.1)
-4.9 t² + 27.5t = 0
Solve the quadratic equation,
t = 5.61224 sec
Therefore, the time it takes for the cannonball to strike the ground if a certain cannon is fired from a height of 8.1 meters above the ground, at a certain angle, the height of the cannonball above the ground, h, in meters, at the time, t, in seconds, is found by the function h(t) =-4.9 t² + 27.5t + 8.1 is 5.612 seconds.
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How do you solve 16/48 = n/36?
Answers
Answer:
n = 12
Step-by-step explanation:
16/48 = n/36
cross multiply
16 × 36 = 48n
576 = 48n
n = 576/48
n = 12
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2. Select all quantities that are proportional to the diagonal length of a square.
A. Area of the square
(B. Perimeter of the square
C. Side length of the square
Answers
For each equation, determine whether it is linear.
(a) y=x
(b) y=-9
Equation
(c) y=4x²¹ +8
(d) y=-4x+7
Is the equation linear?
Yes
No
O
O
X
Answers
Answer:
(a) y=x YES
(b) y=-9 YES
(c) y=4x²¹ +8 NO
(d) y=-4x+7 YES
Step-by-step explanation:
See attached worksheet.
1. f(x)=14(x-3)+1. g(x)=7(x+2)+1
2. f(x)=6(x+2)-3. g(x)=8(x-3)+6
3. f(x)=3(x-1)+²/3. g(x)=3(3-x)-2/3
4. f(x)=-1/3(x+2)-4. g(x)=¹/3(-x-2)+6
5. f(x)=4/3(x-2/3)+¹/3. g(x)=2/3 (2²/3 -x)
State the transformations that occur between f(x) and g(x). I would greatly appreciate if someone could help me with these! Thank you
Answers
Answer:
Comparing the functions, from the tables, it is found that (f - g)(x) is positive in the interval (–∞, 9) .
----------------------
For the subtraction function, we simply subtract both functions, thus:
It is positive if f is greater than g, that is: f(x) > g(x).
It is a linear function, so one function is greater before the equality, one after.
They are equal at x = 9.
If x < 9, f(x) > g(x), and thus, (f - g)(x) is positive, which means that the desired interval is:
(–∞,
Step-by-step explanation:
