Considering the definition of perpendicular line, the equation of the perpendicular line is y= -1/2x + 1/2.
Linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and represents the coordinate of the point where the line crosses the y axis.Perpendicular linePerpendicular lines are lines that intersect at 90° angles. If you multiply the slopes of two perpendicular lines, you get –1.
Equation of perpendicular line in this caseThe line is y= 2x -3
where:
The slope m is 2.The ordinate to the origin b is -3.If you multiply the slopes of two perpendicular lines, you get –1, you get:
2× slope perpendicular line= -1
slope perpendicular line= (-1)÷ 2
slope perpendicular line= -1/2
The line passes through the point (-5, 3). Replacing the point and the slope in the expression for a line:
3= (-1/2)× (-5) + b
3= 5/2 + b
3 -5/2 = b
1/2= b
Finally, the perpendicular line is y= -1/2x + 1/2.
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The diagram below shows triangle OAG where RN, NE, and ER are mid segments of the triangle.
Answer:
here u go
Step-by-step explanation:
The diagram below shows ΔOAG, where RN¯, NE¯, and ER¯ are mid-segments of the triangle.
If OA=(4x+3) units, NE=(2.5x) units, RN=4.5 units, and ER=(2x) units, determine the measures below.
The length of NE¯ is units.
The length of GO¯ is units.
The length of AG¯ is units.
The perimeter of ΔOAG is units.
which statement correctly describes the relationship between △def and △d′e′f′?
△DEF is congruent to △D'E'F' because you can map △DEF to △D'E'F' using a reflection across the x-axis, which is a rigid motion.
What does a math congruent mean?
Congruent refers to having precisely the same form and size. Even after the shapes have been flipped, turned, or rotated, the shape and size ought to remain constant.
1) Reflections, rotations and translations are rigid transformations, because they do not modify the lengths of the segments nor the angles, so the images and the preimages are congruents.
2) Let's see what transformation map △DEF is to △D'E'F' by analyzing the vertices of preimage and image:
Preimage Image
D (-3, -1) D' (-3, 1)
E (2, -4) E' (2, 4)
F (4, -4) F' (4, 4)
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Use Green's Theorem to calculate the circulation of G around the curve, oriented counterclockwise. G = 7yi + xyj around the circle of radius 6 centered at the origin. Close int C G. dr= [ ]
The circle's perimeter is determined by F = -9π, according to green's theorem.
What is Green's theorem?The simple formula for Green's theorem is the link between the total amount of microscopic circulation inside curve C and the macroscopic circulation revolving around it.
Remembering that we are talking about two dimensions, if C is a straightforward closed curve in the plane, it will enclose the region D (shown in red) in the plane.
So, knowing that this exercise should be solved using Green's theorem, then:
[tex]\begin{aligned}& \int 4 y d x+2 x y d y=\iint(d / d x)(2 x y)-(d / d y)(4 y) d A \\& =\iint 2 y-4 d A\end{aligned}[/tex]
You must convert in this manner to polar coordinates, and the result will be:
[tex]\begin{aligned}& \int_0^3 \int_0^{2 \pi}(2 r \sin (\theta)-4)(r) d \theta d r \\& =\int_0^3(-2 r \cos (\theta)-4 \theta)(-2 r \pi) d r \\& =-(\pi)\left(r^2\right)=-9 \pi\end{aligned}[/tex]
Therefore, the circle's perimeter is determined by F = -9π, according to green's theorem.
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Correct question;
Suppose F⃗ (x,y)=4yi⃗ +2xyj⃗ . Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise.
Lola is making greeting cards, which she will sell by the box at an arts fair. She paid $50 for a booth at the fair, and the materials for each box of cards cost $8. She will sell the cards for $10 per box of cards. At some point, she will sell enough cards so that her sales cover her expenditures. How many cards will that take?
At some point, she will sell enough cards so that her sales cover her expenditures. So, the correct answer is ($40)
How did we figure this out?Lola is making greeting cards, which she will sell by the box at an arts fair. She paid $50 for a booth at the fair, and the materials for each box of cards cost $8. She will sell the cards for $10 per box of cards.
We are going to use 50, 8 and 10 to find are answer.
Therefore, we are going to divide the numbers:
[tex]\boxed{50/10}[/tex]
Division problem50/108 x ? = 40What is the missing number?First, we need to figure out 50/10:
[tex]\boxed{50/10=5}[/tex]
[tex]\boxed{8x5=40}[/tex]
[tex]5=40\\(50/10=5=8x5=40)[/tex]
Therefore, at some point, she will sell enough cards so that her sales cover her expenditures. So, the correct answer is ($40)
Answer:
40
Step-by-step explanation:
Let f(x) be a function that is differentiable everywhere and has a derivative f′(x)=4x^2−4x+2. Verify that the Intermediate Value Theorem for Derivatives applies to the function f′(x) on the interval [0,2], and find the value of c guaranteed by the theorem such that f′(c)=5.
The value c guaranteed by the Intermediate Value Theorem over the interval [0, 2], where f(0) = 2, and f(2) = 10 is c = 1.5, f(1.5) = 5
The inequalities, 0 < 1.5 < 2, f(0) < f(1.5) < f2), verifies the Intermediate Value Theorem.
What is the Intermediate Value Theorem?The Intermediate value theorem states that a value p that is between the f(a) and f(b), such that f(a) < p < f(b) or f(a) > p > f(b), then a number, c, exists between a and b such that f(c) = p
The derivative of the function f(x), f'(x) = 4·x² - 4·x + 2
The interval over which the Intermediate Value Theorem is to be verified = [0, 2]
The value of the function at the boundaries of the domain [0, 2], are;
f'(0) = 4 × 0² - 4 × 0 + 2 = 2
f'(2) = 4 × 2² - 4 × 2 + 2 = 10
Therefore, the value, f(c) = 5, indicates that we get;
f'(0) < f(c) < f'(2), and 0 < c < 2
We get; 4·x² - 4·x + 2 = 5
4·x² - 4·x + 2 - 5 = 0
4·x² - 4·x - 3 = 0
Completing the square
4·x² - 4·x = 3
x² - x = 3/4
x² - x + (-1/2)² = 3/4 + (-1/2)²
(x - 1/2)² = 3/4 + (-1/2)² = 1
x - 1/2 = ± √1 = ±1
x = 1 + (1/2) = 1.5, and x = -1 + 1/2 = -1/2
A value of x that satisfies the equation, f'(x) = 4·x² - 4·x + 2, is x = 1.5
0 < 1.5 < 2, when f(0) < f(1.5) < f(2), therefore, the Intermediate Value Theorem is verified on the interval [0, 2] for the derivative, f'(x) = 4·x² - 4·x + 2Learn more about the Intermediate Value Theorem here:
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Find the two numbers which on multiplication with √360 gives a rational number. Are these numbers rational or irrational?
Answer:
The two numbers we are looking for are a rational number and an irrational number.
Step-by-step explanation:
To find the two numbers that, when multiplied by the square root of 360, give a rational number, we can consider the prime factorization of 360. 360 can be written as 2^3 * 3^2 * 5, so the square root of 360 is equal to √(2^3 * 3^2 * 5) = 2√(3 * 5) = 2√15.
Now, suppose we want to find a rational number that is equal to x * √15. We can rewrite this expression as x * √(3 * 5) = x * √3 * √5. For this expression to be rational, both x and √3 must be rational or both must be irrational.
If x is rational, then the expression x * √3 * √5 is rational. Therefore, one of the two numbers we are looking for is x, which is a rational number.
If x is irrational, then the expression x * √3 * √5 is irrational. Therefore, the other number we are looking for is √3, which is an irrational number.
Therefore, the two numbers we are looking for are a rational number and an irrational number.
homework due now!!!!!!!!!!!
Answer:
Option D: 7/3x
Step-by-step explanation:
Because this line passes through the origin (0,0), we know that there is know b-value according to the equation y = mx + b. To figure out the equation, we must first calculate the slope of the linear function.
Slope = △y/△x
△y (change in y) = -7
△x (change in x) = -3
Therefore, the slope is 7/3 and the equation is y = 7/3x.
Evaluate.
12÷(2+23)2
Responses
649
6 and 4 over 9
412
4 and 1 half
323
3 and 2 over 3
1 11/6
The evaluation of the given fraction expression is 1 11/16. The correct option is the last option
Evaluating a fraction expressionFrom the question, we are to evaluate the given expression.
The given expression is
[tex]12 \div (2+\frac{2}{3})^{2}[/tex]
Evaluating the expression
[tex]12 \div (2+\frac{2}{3})^{2}[/tex]
[tex]12 \div (\frac{8}{3})^{2}[/tex]
[tex]12 \div (\frac{8^{2}}{3^{2}})[/tex]
[tex]12 \div \frac{64}{9}[/tex]
= [tex]12 \times \frac{9}{64}[/tex]
= [tex]\frac{108}{64}[/tex]
= [tex]\frac{27 \times 4}{16 \times 4}[/tex]
= [tex]\frac{27}{16}[/tex]
= [tex]1\frac{11}{16}[/tex]
Hence, the fraction is 1 11/16
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one of the factors of g2 − 28g 196 is (1 point) g − 7 g − 14 g 14 g 7
( g - 14)² is factor of equation .
What is factor ?
A number that divides another number by itself with no residual is known as a factor.The factors of 8 are the numbers that divide 8 exactly without leaving any remainder. There are four factors of 8, they are 1, 2, 4, and 8.
In other words, if multiplying two whole numbers results in a product, the numbers we are multiplying are factors of the product since the product is divisible by them. Since the numbers we are multiplying are divisible by the product, they are factors of the product.
g2 − 28g + 196
= g² - 14g - 14g + 196
= g( g- 14 ) - 14( g - 14 )
= ( g - 14) ( g - 14)
= ( g - 14)²
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The complete question is -
One of the factors of g2 -28g + 196 is a. g-7 b. g-14 c. g+14 d. g+7
what is arithmetic sequence
Answer: An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.
The constant difference in all pairs of consecutive or successive numbers in a sequence is called the common difference, denoted by the letter d.
Step-by-step explanation: Hope this was helpful
An arithmetic sequence (arithmetic progression) is a number sequence with a common difference between successive terms. By using the arithmetic sequence formula, we can easily find the value of a term and the common difference in the sequence.
Example: 5, 7, 9, 11, 13, 15
What will be the eauation of vertical asymptote x=1 and x=5 and horizontal asymptote y=0. The graph does not have x intercept and it passess (4,-2)
Answer:
The figure below shows the graph of rational function f; It has vertical asymptotes x 2 andx = 4, and horizontal asymptote y = 0. The graph does not have an x-intercept, and it passes through the point (~3,-1). The equation for f (x) has one of the five forms shown below. Choose the appropriate form for f (x), and then write the equation: You can assume that f (x) is in simplest form_ 0 f() f() f ()
what is the least common denominator of 3/5 and 1/2
Answer:10
Rewriting input as fractions if necessary:
3/5, 1/2
For the denominators (5, 2) the least common multiple (LCM) is 10.
LCM(5, 2)
Therefore, the least common denominator (LCD) is 10.
Calculations to rewrite the original inputs as equivalent fractions with the LCD:
3/5 = 3/5 × 2/2 = 6/10
1/2 = 1/2 × 5/5 = 5/10
Answer:
Step-by-step explanation:
1/2 + 3/5 = 5/10 + 6/10 = (5+6)/10 = 11/10 or 1 1/10 Your pair of fractions is 5/10 and 6/10
1. Business is major
The sampling distribution of p
PROBLEM:
According to the National Center for Education Statistics, 20% of students who
recently earned bachelor's degrees were business majors. Imagine taking an SRS of
300 students who recently earned bachelor's degrees and calculating p = the
proportion of students in the sample who majored in business.
(a) Identify the mean of the sampling distribution of p.
(b) Calculate and interpret the standard deviation of the sampling distribution of p.
Verify that the 10% condition is met.
(c) Describe the shape of the sampling distribution of p. Justify your answer.
a) The mean of the sampling distribution of p is of 0.2.
b) The standard deviation of the sampling distribution of p is of: 0.0231, which is the amount that the sample proportions may differ from 0.2.
c) The shape of the sampling distribution of p is approximately normal.
How to define the sampling distribution of p?The sampling distribution of p is defined according to the Central Limit Theorem, which states that the sampling distribution of a proportion p in a sample of size n has:
Approximately normal shape.Mean equals to p.Standard deviation equals to [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex]The parameters for this problem are given as follows:
p = 0.2, n = 300.
The 10% condition is given as follows:
np = 300 x 0.2 = 60 > 10.n(1 - p) = 300 x 0.8 = 240 > 10.Then the standard error of the sampling distribution of p is calculated as follows:
[tex]s = \sqrt{\frac{0.2(0.8)}{300}} = 0.0231[/tex]
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The line ax+by+c = 0 passes through the points (2 ,3) and (8 , 15). Find the ratio a :b :c.
The ratio of a : b : c in the linear equation is 2 : -1 : -1
How to determine the ratio of a : b : c?From the question, we have the following equation that can be used in our computation:
ax + by + c = 0
The points on the line are given as
(2 ,3) and (8 , 15)
A linear equation is represented as
y = mx + c
Where
Slope = m
y-intercept = c
This means that
3 = 2m + c
15 = 8m + c
Subtract the equations
So, we have the following representation
6m = 12
Divide by 6
m = 2
Substitute m = 2 in 3 = 2m + c
3 = 2 * 2 + c
So, we have
c = -1
Substitute m = 2 and c = -1 in y = mx + c
y = 2x - 1
So, we have
2x - y - 1 = 0
Recall that
ax + by + c = 0
This means that
a : b : c = 2 : -1 : -1
Hence, the ratio is 2 : -1 : -1
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Give an example of each of the following: (a) A natural number (b) An integer that is not a natural number (c) A rational number that is not an integer (d) An irrational number
a) Natural number: [tex]12[/tex]
b) Integer that is not a natural number:[tex]-8[/tex]
c) Rational number that is not an integer: [tex]\frac{5}{10} or 0.5[/tex]
d) An Irrational number: [tex]\sqrt{3}[/tex]
A natural number is any whole number greater than zero, such as [tex]1,2,3,4,5,6[/tex] etc. Natural numbers are also known as counting numbers.
An integer is a whole number (not a fraction) that can be positive, negative, or [tex]0[/tex]. Examples of integers are: [tex]-5,0,1,5,10.[/tex]
Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are integers. Examples of rational numbers include [tex]-\frac{1}{2},\frac{3}{4},0.75, etc.[/tex]
Irrational numbers are numbers that cannot be expressed as a simple fraction and that cannot be written as a repeating or terminating decimal. Examples of irrational numbers include [tex]\pi (3.14), \sqrt{2}, etc.[/tex]
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a) Natural number: 12
b) Integer that is not a natural number:-8
c) Rational number that is not an integer: [tex]\frac{5}{10}[/tex]
d) An Irrational number: [tex]\sqrt{5}[/tex]
A natural number is any whole number greater than zero, such as [tex]1,2,3,4,5[/tex] etc. Natural numbers are also known as counting numbers.
An integer is a whole number (not a fraction) that can be positive, negative, or 0. Examples of integers are: [tex]-1,5, etc[/tex]
Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are integers. Examples of rational numbers include [tex]\frac{1}{2} ,0.5[/tex]
Irrational numbers are numbers that cannot be expressed as a simple fraction and that cannot be written as a repeating or terminating decimal. Examples of irrational numbers include [tex]\pi ,\sqrt{6}[/tex]
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Lincoln decides to research the relationship between the length in inches and the weight of a certain species of catfish. He measures the length and weight of a number of specimens he catches then throws back into the water. After plotting all his data, he draws a line of best fit. What is the meaning of the xx-value on the line when y=29y=29?
The xx-value on the line when y=29 represents the length in inches of the catfish that would have a weight of 29.
Why are you talking about measurement?Measuring is the act or process of determining the connection between two numbers. The act of measuring is the practice of putting a number on a characteristic of an object or event so that it can be compared to other things and phenomena. Calculating something's size or magnitude is what it is.
What purpose does measurement serve?Comparing unknown amounts to known quantities is made easier by measurement. Measurement enables us to establish quantitative claims about the size, length, and speed of objects. Without measurement, the result will be inaccurate. Examples: Vehicle speed is measured using a speedometer.
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Car A travels 702 miles on 12 gallons of gasoline. Car B travels 873 miles on 15 gallons of gasoline. Hugo wants to buy a car with lower fuel consumption. Find out the distance each car could travel per gallon of gasoline. Then, tell which of the two cars (A or B) Hugo should buy.
Answer: Hugo should buy car A
Step-by-step explanation:
Car A
Take 702 divided by 12 = 58.5 miles per gallon of gas
Car B
Take 873 divided by 15 = 58.2 miles per gallon of gas.
So Hugo should buy car A
Name Natalie Schafer
6) I bought 6 pieces of candy. I paid $0.40 per Tootsie Roll and $0.15 per
Dubble Bubble for a total of $1.90. How many Tootsie Rolls and Dubble
Bubbles did I buy ?
Answer: 6 pieces of candy
Step-by-step explanation:
The lifting force, F, exerted on an airplane wing varies jointly as the area, A, of the wing's surface and the square of the plane's velocity, v. The lift of a wing with an area of 250 square feet is 30,300 pounds when the plane is going at 190 miles per hour. Find the lifting force if the speed is 180 miles per hour. Round your answer to the nearest integer if necessary.
F= k*a*v^2
a= 250
v=190
30300=k*250*{190^2}
k= 0.003357
F= 0.003357*250*180*180
F=27135
The lifting force is 27135
What is surface area and an illustration?The total area that all of a 3D object's faces cover is the surface area. The surface area of a cube is its surface area, for instance, if we are trying to determine how much paint is needed to paint it. Always expressed in square units.
How does one determine surface area?The entire surface of a three-dimensional shape is referred to as its surface area. A cuboid with six rectangular faces has a surface area equal to the sum of the areas of each face. Alternatively, you can label the cuboid's length, width, and height (l, w, and h), then calculate its surface area (SA) using the formula: SA=2lw+2lh+2hw.
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The graph of the equation y = 3x + 1 is shown below.
If the graph is reflected across the y-axis, what will be the equation of the new graph?
Answer:
y = -3x + 1
Step-by-step explanation:
Mr. Callaway needs to purchase enough grass seed to cover a 3000-square-foot lawn and a
4200-square-foot lawn. If 40 ounces of grass seed will seed a 2400-square-foot lawn, how many ounces
does he need to seed both lawns?
Answer:
Mr. Callaway needs 120 ounces of seed to seed both lawns
Step-by-step explanation:
2400/40 = 60 ounces per square foot
3000 divided by 60 = 50 ounces
So, Mr. Callaway needs 50 ounces to cover a 3000-square-foot lawn
4200 divided by 60 = 70 ounces
So, Mr. Callaway needs 70 ounces to cover a 4200-square-foot lawn
50+ 70 = 120 ounces
So, Mr. Callaway needs 120 ounces to seed both lawns.
If you can, please give me a Brainliest; thank you, and have a good day!
Answer:
120 ounces pf grass seed
Step-by-step explanation:
[tex]\frac{40}{2400}[/tex] = [tex]\frac{x}{3000}[/tex] Cross multiply and solve for x
120000 = 2400x Divide both sides by 2400
\[tex]\frac{120000}{2400}[/tex] = [tex]\frac{2400x}{2400}[/tex]
50 = x
It will take 50 ounces of grass seed to cover 3,000 square foot lawn.
[tex]\frac{40}{2400}[/tex] = [tex]\frac{x}{4200}[/tex] Cross multiply and solve for x
168000 = 2400x Divide both sides by 2400
70 = x
It will take 70 ounces of grass seed to cover 4200 square foot lawn.
50 + 70 = 120
50 POINTS! Please Help!
James is buying furniture for her new apartment. She buys one dining room table and four chairs and cost $265. Mark buys 2 tables and 6 chairs for the total cost of $470.
The system of equations below represents the scenario.
x + 4y = 265
2x + 6y = 470
Define the variables for x and y.
Determine the cost for one dining room table.
Using the system of equation, the cost of one dinning table is 145 dollars.
How to find the cost of one dinning table?James is buying furniture for her new apartment. She buys one dining room table and four chairs and cost $265. Mark buys 2 tables and 6 chairs for the total cost of $470.
The system of equation that represent the situation is as follows:
let
x = cost of one dinning room table
y = cost of one chair
Therefore,
x + 4y = 265
2x + 6y = 470
using substitution method,
x = 265 - 4y
2(265 - 4y) + 6y = 470
530 - 8y + 6y = 470
-2y = 470 - 530
-2y = -60
y = -60 / -2
y = 30
Hence,
x = 265 - 4(30)
x = 265 - 120
x = 145
Therefore,
cost of one dinning table = 145 dollars
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b(x) = 6x², when x = 16
Answer:
Step-by-step explanation:
b(x)=6x^2
That is the same as b(x)=6*x*x
when x=16
b(16)=6*16*16=1536
For What Values Of X And Y Are The Triangles To The Fight Congruent By HL? X = And Y =
The values of x and y the triangles to the fight congruent are x = 3 and
y = 1.
What is congruent triangles?
Triangles with the same size and shape are said to be congruent. This implies that the corresponding sides and angles are both equal. Without comparing every angle and side of the two triangles, we can determine whether two triangles are congruent.
If given these triangles are congruent then all corresponding side must be equal
As we can see that both triangle are right triangle
Therefore hypotenuse of one triangle equal to another
x+1 = 4y ..... (A)
And lenght ( height) of one must equal to another triangle
x = y +2 .... (B)
therefore
Plug the value of x = y + 2 in equation A
y + 2 + 1 = 4y
3y = 3
y = 1
and x = y+2
x = 1 + 2
x = 3
Hence, the values of x and y the triangles to the fight congruent are
x = 3 and y = 1.
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Complete question:
Complete question is attached below.
henry wants to earn at least trimming trees. he charges per hour and pays in equipment fees. what are the possible numbers of hours henry could trim trees?
The possible number of hours Henry could trim trees is t > 10.
The given information represents a scenario which demonstrates an inequality statement as Henry wants to earn at least $ 56, which means the amount earned should be greater than 56.
Let t be the possible number of hours Henry trim trees. As his per hour charge is $6 and he pays $4 in equipment fee, the inequality statement is:
6t – 4 > 56
Solving,
6t > 56 + 4
6t > 60
t > 10.
In order to earn at least $56, Henry must trim trees for at least 10 hours or more.
Note: The question is incomplete. The complete question probably is: Henry wants to earn more than $56 trimming trees. He charges $6 per hour and pays $4 in equipment rental fees. What are the possible number of hours Henry could trim trees?
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Esb poles are placed along a road 70 metres apart. What is the distance between the 1st and the 3rd poles
The distance between the 1st and the 3rd poles is 140 meters.
What is the arithmetic operation?
The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.
ESB poles are placed along a road 70 meters apart.
So the distance between two poles is 70 meters.
So the distance between 1 and 3rd will be = 2*70 =140metets
Hence, the distance between the 1st and the 3rd poles is 140 meters.
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Complete Question:
Esb poles are placed along a road 70 metres apart. What is the distance between the 1st and the 3rd poles?
the sample midrange for the data. (ii) explain why the sample midrange might be preferred to the sample mean as an estimator of the population mean.
Answer:
The sample midrange for the data is 559
What is Sample midrange?
Although the sample midrange only uses a small fraction of the data, outliers can have a significant impact on it. It offers details on the distribution's skewness and heavy tails, which are identical to those of the random variable's distribution. Although the structure of this distribution is not obvious, with large sample sizes, the central limit will cause it to resemble a normal distribution.
According to question:
Minimum value: 34
Maximum values: 1084
The sample midrange can be computed as:
(Minimum value + Maximum value) / 2
⇒ (34+1084)/2
⇒ (1118)/2
⇒ 559
Hence, the sample midrange is 559
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The Complete Question is
HELPP!!!
angles of a triangle
Math
help please! Thank u.
Answer:
Step-by-step explanation:
40/20= 20
39/x=x
that woudl be how you would find it
Find the derivative please
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]
