Yes, both methods give the same result. The chain rule is used to differentiate a composite function, while the substitution method is used to differentiate a function after substituting a variable with its expression.
The chain rule is used to differentiate a composite function, which is a function of a function. To use the chain rule, we differentiate the inner function and multiply it by the derivative of the outer function. In this case, we would have
f(x) = ([tex]x^2 + 3x + 2[/tex]) and
g(x) = [tex]2x^3 + 5[/tex]
The derivative of f(x) would be 2x + 3 and the derivative of g(x) would be [tex]6x^2[/tex]. Taking the product of these two derivatives, we get
[tex]12x^3 + 18x^2[/tex],
which is the answer we get using the chain rule. The substitution method involves substituting a variable in a function with its expression. In this case, we substitute x with ([tex]x^2 + 3x + 2[/tex]). This gives us a new function that is a function of [tex]x^2 + 3x + 2[/tex]instead of x. We then differentiate this new function and the answer we get is the same as the one we got using the chain rule. Therefore, both methods give the same result and agree with each other.
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A class has 45 students of the students are wearing blue shirts, of
them are wearing red shirts, and the rest are wearing yellow shirts.
What is the probability of selecting a student who is wearing a yellow
shirt?
The probability of selecting a student who is wearing a yellow shirt is 36 %
How to find the probability ?The total number of students is 45 students and out of these, 3 / 7 are wearing blue shirts while 2 / 9 are wearing red shirts.
This means that the number of students wearing yellow is:
= 45 - ( 45 x 3 / 7 ) - ( 45 x 2 / 9 )
= 45 - 19 - 10
= 16 students
The probability of selecting a student who is wearing yellow is :
= 16 / 45
= 36 %
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The full question is:
A class has 45 students of the students are wearing 3/7 blue shirts, of
them are 2/9 wearing red shirts, and the rest are wearing yellow shirts.
What is the probability of selecting a student who is wearing a yellow
shirt?
The function f(x) represents the number of texts sent this week, and g(x) represents the number of texts sent last week. Given f(x) = 7x + 5 and g(x) = 6x − 8, find (f − g)(x) to find how many more texts were sent this week than last week.
(f − g)(x) = x + 13
(f − g)(x) = x − 13
(f − g)(x) = 13x + 13
(f − g)(x) = 13x −13
Answer:
Step-by-step explanation:
(f - g) (x) = f(x) - g(x)
Given that f(x) = 7x + 5 and g(x) = 6x - 8, we can substitute these values into (f - g) (x):
(f - g) (x) = (7x + 5) - (6x - 8)
(f - g) (x) = 7x + 5 - 6x + 8
(f - g) (x) = x + 13
So, the answer is (f - g) (x) = x + 13
To find how many more texts were sent this week than last week we use,
(f − g)(x) = x + 13.
What is a function?A function can be defined as the outputs for a given set of inputs.
The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.
From the given information, The number of more texts sent this week than last week is the difference between the functions f(x) - g(x).
= (f - g)x.
= (7x + 5) - (6x - 8).
= 7x + 5 - 6x + 8.
= x + 13.
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x – 0.30x = 0.70x
A - Decrease a number by 30% is the same as multiplying by 70%.
B - Decrease a number by 30% is the same as increasing by 70%.
C - Decrease a number by 0.30 is the same as multiplying by 0.70.
D - Decrease a number by 30 is the same as multiplying by 70.
Answer:
A - Decrease a number by 30% is the same as multiplying by 70%.
Step-by-step explanation:
This statement is incorrect. Decreasing a number by 30% means that the number will be reduced by 30% of its original value. For example, if the original number is 100, decreasing it by 30% would give you 100 - 30% * 100 = 100 - 30 = 70. On the other hand, multiplying a number by 70% would increase it by 70% of its original value, which is a different operation.
C - Decrease a number by 0.30 is the same as multiplying by 0.70.
This statement is also incorrect. Decreasing a number by 0.30 means subtracting 0.30 from the original number, while multiplying by 0.70 means multiplying the original number by 0.70. These are two different operations that will result in different results.
a study shows that at the end of a school day teenagers typically have 40% of their cell phone battery left when plugged in phone charge at a rate of 3% per minute when using the newest fast charger.
a: write an inequality to represent how many more minutes are needed to reach at least 80% battery life.
b. use the inequality to determine the how many more minutes it will take to have at least 80% battery life
Part(a), An inequality to show the number of additional minutes required to get at least 80% battery life will be 3m + 40 ≤ 80.
Part(b) The number of minutes will be 13 minutes and 20 seconds.
What is inequality?The inequality expressions are the mathematical equations related by each other by using the signs of greater than or less than. All the variables and numbers can be used to make the equation of inequality.
Given that according to a survey, students often have 40% of their cell phone batteries left at the end of the school day when utilizing the newest fast charger, which charges phones at a pace of 3% per minute.
Part(a) The inequality can be written as:-
3m + 40 ≤ 80.
Part(b), The number of minutes will be:-
3m + 40 ≤ 80.
3m ≤ 40
m ≤ 40 / 3
m ≤ 13.33
The time can also be written as 13 minutes and 20 seconds.
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Isabella is conducting an experiment with a 12 sided dice. The sample space is 1 through 12. What is the percent probability that Isabella will roll a 14?
Responses
20
20
0
0
10
10
15
The percent probability that Isabella will roll a 14 is 0%, since the dice she is using has a sample space of 1 through 12 and there is no 14 on the dice. Therefore, the probability of her rolling a 14 is 0%.
The probability of Isabella rolling a 14 with a 12-sided dice is 0%. This is because the sample space of the dice is 1 through 12, and there is no 14 on the dice. In order to calculate the probability, you would need to take the number of possible outcomes (the sample space) and divide it by the number of outcomes you are looking for. In this case, the sample space is 12 and the outcome being searched for is 1, giving you 12 divided by 1 which is 0. This means that the probability of Isabella rolling a 14 is 0%. This is because the sample space does not contain a 14, thus making it impossible for Isabella to roll a 14.
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Find the greatest whole number which divides 372 & 324 without leaving remainder
The greatest whole number that divides both 372 and 324 without a remainder is 4.
To find the greatest whole number that divides both 372 and 324 without a remainder, we can use the concept of the greatest common divisor (GCD).
One method to find the GCD is to factor each number into its prime factors and then find the product of the common prime factors.
372 can be factorized as 2 * 186, and 324 can be factorized as [tex]2^2 * 3^4[/tex]
The greatest common divisor is [tex]2^2[/tex], which is equal to 4.
So, the greatest whole number that divides both 372 and 324 without a remainder is 4. This means that 4 is the largest number that can divide both 372 and 324 evenly, leaving no remainder.
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[tex]64^{2x+1} =1024[/tex]
Answer:
x = 1/3
Step-by-step explanation:
64^2x+1 = 1024
4^3(2x+1) = 4^5
Since the bases are the same, two expressions are only equal if the exponents are also equal.
3 (2x + 1) = 5
Divided each term by 3
2x + 1 = 5/3
2x = 2/3
x = 1/3
Find the distance of a chord of Length 16cm From the centre of a Circle of radius 12.Cm.
The distance of the chord from the center of the circle is derived to be equal to √80 cm or 8.9 cm to the nearest tenth using the Pythagoras rule.
What is the Pythagoras rule?The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The the distance can be observed to be of the side of a right triangle with the hypotenuse as the radius of the circle and the base side half of the chord, if x is used to represent the distance of the chord from the centre of the circle.
so by Pythagoras rule we can write:
12² = x² + 8²
144 = x² + 64
x² = 144 - 64 {collect like terms}
x² = 80
x = √80 {take square root of both sides}
x = 8.9443
Therefore, the distance of the chord from the center of the circle is derived to be equal to √80 cm or 8.9 cm to the nearest tenth using the Pythagoras rule.
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Suppose normal distribution has a mean of 98 and a standard deviation of
6. What is P(x ≥ 92)?
O A. 0.16
B. 0.475
C. 0.84
D. 0.975
The probability expressed as P(x ≥ 92) in the normal distribution is; C: 0.84
How to find the probability in normal distribution?We are given that the normal distribution has;
Population mean: μ = 98
Population standard deviation; σ = 6
Sample mean; x' = 92
Thus, z-score is;
z = (x' - μ)/σ
z = (92 - 98)/6
z = -1
Thus, from z-score tables we have;
P(z ≥ -1) = 0.84
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The height of a cone is 10 centimeters. The radius is 4 centimeters. What is the volume of this cone? Around your answer to the nearest hundredth. Do not include units in your answer.
YOUR MISSION: Create and solve a linear equation based on a real-world problem on any media of your choice (e.g. poster board, computer, etc). Make sure you only have 1 variable (let's say X), but this variable can be used multiple times throughout the equation. For example, 2(X + 1) = 3X + 5-1X
Note: A constant is a number that is standing all by itself as its own term. For example, in 2X+1=5, the constants are 1 and 5. A coefficient is the number attached to the left of the variable, also known as in front of the variable. For example, in 2X +1 =5, the coefficient is 2.
Fill in the table using this function rule.
y = 4x+2
X
-2
0
2
4
y
0
0
0
0
X
D
The table is completed using this function rule y = 4x + 2 is given below.
x -2 0 2 4
y -8 2 10 18
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
The function is given below.
y = 4x + 2
At x = - 2, the value of 'y' is given as,
y = 4 × (-2) + 2
y = - 8 + 2
y = - 6
At x = 0, the value of 'y' is given as,
y = 4 × (0) + 2
y = 0 + 2
y = 2
At x = 2, the value of 'y' is given as,
y = 4 × (2) + 2
y = 8 + 2
y = 10
At x = 4, the value of 'y' is given as,
y = 4 × (4) + 2
y = 16 + 2
y = 18
The table is completed using this function rule y = 4x + 2 is given below.
x -2 0 2 4
y -8 2 10 18
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Step-by-step explanation:
Please here it is.
Given y = 4x + 2
There are 12 girls and 14 boys in math class. The teacher puts the names of the students in a hat and randomly picks one name. Then the teacher picks another name without replacing the first.
What is the probability that both students picked are the same gender?
Answer:
The probability of picking the first boy is 14/26 and the probability of picking another boy is 13/25.
Step-by-step explanation:
Pls show your work thank you will mark the Brainliest
The graph that best models the situation of the home value increasing every year is Graph A
How to find the graph ?To find the graph, use the percentage increase to find the value of the house in two different time periods and then check which graph coincides with this prediction.
After 5 years, the value of the house should be:
= 200, 000 x ( 1 + 6 % ) ⁵
= $ 267, 645
After 10 years :
= 200, 000 x ( 1 + 6 % ) ¹⁰
= $ 358, 170
The best graph for the situation is therefore Graph A which starts from $ 200, 000 and passes through the above points.
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Shape A is reflected in the line with equation x=2 to get Shape B. Shape B is reflected in the x=6 to get Shape C. Describe the transformation that fully transforms Shape A to Shape C.
The transformation that fully transforms Shape A to Shape C is reflected in the x =6.5
What is a reflection?There are two ways of reflection.
Along x-axis:
(x, y) – (x, -y)
Along y-axis:
(x, y) - (-x, y)
We have,
The line x = 2, x = 6, and x = 6.5 is given on the graph below.
Shape A is reflected in the line with equation x = 2 to get Shape B.
Shape B is reflected in the x = 6 to get Shape C.
Now,
Shape A is reflected in the x =6.5 to get Shape C.
Thus,
The transformation that fully transforms Shape A to Shape C is reflected in the x =6.5
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can someone please help i don’t understand
36-7p=-7(p-5) solve for P
Step-by-step explanation:
36 - 7p = -7(p-5)
= 36 - 7p = -7p + 35
= 36 - 35 = - 7p + 7p
= 1 = 0
am confused at the last part
I don't know if p= 1 or it 0 = 1
a sample of students will be selected from all the students at a high school. which of the following sampling methods is least likely to produce a representative sample of the students? responses from a list of all student names, randomly select 50 names from the list for the sample. from a list of all student names, randomly select 50 names from the list for the sample. randomly choose five classrooms in the school and use all the students in those classrooms for the sample. randomly choose five classrooms in the school and use all the students in those classrooms for the sample. divide the students into the four class years (freshman, sophomore, junior, senior) and randomly select students from
Divide the students into the four class years (freshman, sophomore, junior, senior) and randomly select students from each year in proportion to the number of students in each year.
Research Sampling Methods:There are several ways to obtain a sample in research. The most common sampling methods in research are the following:
Simple random: A sampling method in which each member of the population has an equal probability of being selected for the sample.Convenience: A sampling method in which the sample is collected from a convenient group or place for the researchers.Systematic: A sampling method in which a sample is selected by choosing every n-th member of the population for some integer n≥2.Stratified: A sampling method in which a sample is selected according to different characteristics of the subjects (i.e., gender, age, disease status).Learn more about Sampling Method at:
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Substitute and simplify.
w = -6, x = 4, y = 3, z = -8
1. WX + YZ =
2. 2w + 3z- xy + z³ =
3. y(z + x) =
4. (W+x+y)³=
5. (z-w)³ =
6. 5(w + x) + 4(y + z) =
7. W² - x² =
8. (xy) – 2wz =
9. wz - 4xy =
10. w+ (x+y+z)² =
The solution to all the Equations is shown below.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
w = -6, x = 4, y = 3, z = -8
1. WX + YZ
= (-6) (4) + (3)(-8)
= -24 - 24
= -48
2. 2w + 3z- xy + z³
= 2(-6) + 3(-8) - (4)(3) + (-8)³
= -12 - 24 - 12 -512
= -560
3. y(z + x)
= 3(-8 + 4)
= -12
4. (w + x + y)³
= (-6 + 4 + 3)³
= 1
5. (z-w)³
= (-8 + 6)³
= -8
6. 5(w + x) + 4(y + z)
= 5(-6 + 4) + 4(3 -8)
= -10 - 20
= -30
7. W² - x² =
= (-6)² - (4)²
= 36 - 16
= 20
8. (xy) – 2wz
= (4)(3) - 2(-6)(-8)
= 12 -2 (48)
= 12 -96
= -84
9. wz - 4xy
= (-6)(-8) - 4(4)(3)
= 0
10. w+ (x+y+z)²
= -6 + 1
= -5.
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Macy wants to join a Rock-Climbing gym. Avery's gym has an initial cost of $50 with a $10 per session charge. Gabe's gym has no initial cost, but has a $12 per session charge. How many sessions will she have to take to make both gym's cost the same?
Answer: 25 sessions
Step-by-step explanation:
Set up an equation.
50+10x=12x
50=2x
x=25
She will have to take 25 sessions for both gym costs to be the same.
You can check this too. 50 plus 10*25 is $300, and 12*25 is $300.
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Hope this helps.
MM4343
Answer: We can start by setting the cost of the two gyms equal to each other and then solving for the number of sessions. Let's call the number of sessions Macy takes x. The cost of Avery's gym is 50 + 10x, and the cost of Gabe's gym is 12x. Setting these two equal to each other, we have:
50 + 10x = 12x
Subtracting 12x from both sides:
50 = -2x + 10x
Simplifying:
50 = 8x
Dividing both sides by 8:
x = 6.25
So, Macy would need to take 6.25 sessions for the cost of both gyms to be the same. However, since she can only take whole number of sessions, she would need to take 7 sessions to make both gyms cost the same.
Step-by-step explanation:
Evaluate (-7)⁰
What’s is the solution?
Answer:
1
Step-by-step explanation:
everything to power of 0 is 1, except for 0^0 is undefined
Uriel collects aluminum cans, crushes them with a hand can-crushing machine, and turns the crushed cans in once a month for extra money. Uriel can crush 3 cans in 23 of a minute. What is the rate, in cans per minute, at which Uriel crushes aluminum cans?
If Uriel can crush 3 cans in 23 of a minute, Uriel crushes aluminum cans at a rate of 4.5 cans per minute.
The problem gives us information about the number of cans Uriel can crush and the time it takes him to do so. We want to find the rate at which he crushes the cans, which is a measure of how many cans he can crush per minute.
To find this rate, we use a proportion. A proportion is a statement that two ratios are equal. In this case, we can set up a proportion that relates the number of cans Uriel crushes to the time it takes him to do so, and the rate at which he crushes the cans:
3 cans / (2/3 min) = x cans / 1 min
Here, the first ratio is the number of cans Uriel can crush in 2/3 of a minute (which is the same as saying that he can crush 3 cans in 2/3 of a minute). The second ratio is the unknown rate at which Uriel crushes cans per minute.
To solve for the unknown rate, we use cross-multiplication. This involves multiplying both sides of the proportion by the same value in order to isolate the unknown variable on one side of the equation.
In this case, we can cross-multiply by the denominators of each ratio:
3 cans × 1 min = (2/3 min) × x cans
Simplifying the left-hand side, we get:
3 cans × 1 min = 3 cans/min
Simplifying the right-hand side, we get:
(2/3 min) × x cans = 2x/3 cans/min
Putting these together, we get the equation:
3 cans/min = (2x/3) cans/min
We can now solve for x by isolating it on one side of the equation. To do this, we can multiply both sides by 3/2:
(3/2) × 3 cans/min = x cans/min
Simplifying, we get:
4.5 cans/min = x cans/min
Therefore, Uriel crushes aluminum cans at a rate of 4.5 cans per minute.
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Please help with this homework ?
Step-by-step explanation:
sorry for the wait I hope this helps you with you're problem
question 14
help me asap
Answer:
arc PN = 164°
Step-by-step explanation:
given chords PM and PN are congruent.
• if 2 chords are congruent then the corresponding arcs are congruent
then
arc PN = arc PM = 164°
part b. which equation has the same graph as the answer from part a, and so can also be used to model the data?
Answer:
D
Step-by-step explanation:
f(x)=20 cos (pi/5x + pi/2)+25
Use a trigonometric ratio to solve for z. Round to two decimal places as necessary.
The value of z in the triangle using trigonometric ratio is; z = 10.90
How to use trigonometric ratios?The three basic trigonometric ratios from which others are derived are;
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
From the diagram, we can easily see that z is the opposite side of the given angle and 13 is the hypotenuse. Thus;
z/13 = sin 57
z = 13 * sin 57
z = 13 * 0.83867
z = 10.90
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can you answer the question in the picture and show me the steps
Step-by-step explanation:
Since it varies directly, we can set up a ratio and solve for x
[tex] \frac{ - 4}{ - 2} = \frac{x}{12} [/tex]
Next we would cross multiply... multiply the numbers diagonally
[tex]( - 4)(12) = ( - 2)(x)[/tex]
Simplifying
[tex] - 48 = - 2x[/tex]
Now to get X alone, we would divide by the number attached to it, so -2. We'd do -2 on the -48 also
[tex]x = 24[/tex]
Use the place value to solve the following 92-12=
Answer:
hey there here's your answer !!!!
Step-by-step explanation:
For two-digit numbers, there is the TENS place and the ONES place. Let's take a closer look at the place value for the number 12. The 1 is in the TENS place, and the 2 is in the ONES place. That's because the number 12 is made up of 1 ten and 2 ones.
in a particular region during a 1-year period, there were 1000 deaths. it was observed that 321 people died of a renal failure and 460 people had at least one parent with renal failure. of these 460 people, 115 died of renal failure. calculate the probabil ity of a person that he dies of renal failure if neither of his parents had a renal failure
The probability of a person dying of renal failure if neither of their parents had renal failure is 0.056 or about 5.6%.
To solve this problem, we can use Bayes' theorem, which states that:
P(A|B) = P(B|A) * P(A) / P(B)
where P(A|B) is the probability of event A given that event B has occurred, P(B|A) is the probability of event B given that event A has occurred, P(A) is the prior probability of event A, and P(B) is the prior probability of event B.
In this case, we want to find the probability of a person dying of renal failure given that neither of their parents had renal failure. Let's define the following events:
R: death due to renal failure
P: at least one parent with renal failure
NP: neither parent has renal failure
We are given the following information:
Total deaths = 1000
Deaths due to renal failure (R) = 321
People with at least one parent with renal failure (P) = 460
Deaths due to renal failure among people with at least one parent with renal failure = 115
Using this information, we can calculate the probabilities as follows:
P(R) = 321/1000
P(P) = 460/1000
P(R|P) = 115/460
To find the probability of a person dying of renal failure given that neither of their parents had renal failure (i.e., P(R|NP)), we need to use Bayes' theorem. We can start by finding the probability of neither parent having renal failure (i.e., P(NP)):
P(NP) = 1 - P(P) = 1 - 0.46 = 0.54
Next, we can use the law of total probability to find the probability of dying of renal failure among people with neither parent having renal failure:
P(R|NP) = P(R) * P(NP|R) / P(NP)
We can find P(NP|R) using the formula:
P(NP|R) = P(NP ∩ R) / P(R)
To find P(NP ∩ R), we can use the formula:
P(NP ∩ R) = P(R|NP) * P(NP)
Substituting the values, we get:
P(R|NP) = P(R) * P(NP|R) / P(NP)
= 321/1000 * (1 - 115/460) / 0.54
= 0.056
This calculation shows how conditional probabilities can be calculated using Bayes' theorem and the law of total probability.
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Help, I got 2262, I dunno what I'm doing wrong.
The two states will never have the same median house value.
How to model the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
The slope m represents the rate of change.The intercept b represents the value of y when x = 0.The intercepts are given by the values of each home in 1950, thus:
Idaho: 35500.Delaware: 55000.The slopes are given by the average change over the 50 years, hence:
Idaho: m = (106300 - 35500)/50 = 1416.Delaware: m = (130400 - 55000)/50 = 1508.As the number of years must be positive, Delaware starts with the higher income and has a higher yearly increase, the two states will never have the same median house value.
Solving 35500 + 1416x = 55000 + 1508x, we would end up with a negative value of x.
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