Use the Chain Rule to express the second derivative of f∘g in terms of the first and second derivatives of f and g.

Use The Chain Rule To Express The Second Derivative Of Fg In Terms Of The First And Second Derivatives

Answers

Answer 1

The second derivative of the composition between two functions is described by the following expression:

[tex]\frac{d^{2}}{dx^{2}} (f\,\circ\,g\,(x)) = \frac{d^{2}f}{dg^{2}}\cdot \frac{dg}{dx} + \frac{df}{dg} \cdot \frac{d^{2}g}{dx^{2}}[/tex]

Mathematically speaking, a composition between two functions is defined by the following operation:

[tex]f\,\circ\,g\,(x) =f(g(x))[/tex] (1)

By Chain Rule we get the first and second derivatives of the composition:

First derivative

[tex]\frac{d}{dx} (f\,\circ\,g\,(x)) = \frac{df}{dg}\cdot \frac{dg}{dx}[/tex] (2)

Second derivative

[tex]\frac{d^{2}}{dx^{2}} (f\,\circ\,g\,(x)) = \frac{d^{2}f}{dg^{2}}\cdot \frac{dg}{dx} + \frac{df}{dg} \cdot \frac{d^{2}g}{dx^{2}}[/tex] (3)

The second derivative of the composition between two functions is described by the following expression:

[tex]\frac{d^{2}}{dx^{2}} (f\,\circ\,g\,(x)) = \frac{d^{2}f}{dg^{2}}\cdot \frac{dg}{dx} + \frac{df}{dg} \cdot \frac{d^{2}g}{dx^{2}}[/tex]

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Related Questions

1
Write the equation
for this line in
slope-intercept form.
y = -2x + [?]
4-3-4-1
34
op
A 4
B. 2
C -2
D. -12

Answers

Answer:

A, -4 is the answer

Step-by-step explanation:

it intersects there

1. A cylindrical jar has a radius of 5 inches and a height of 9 inches. The jar is filled with marbles that have a volume of 20 in3. Use 3.14 for pi. Show work. Complete sentences.
a. What is the volume of the jar?









b. How many whole marbles can fit in the jar?
who ever gets the right answer will get brainliest ;)

Answers

Answer:

a. Volume = π15.02 9.0=6361.73

b. I am very sorry, I don't know

Step-by-step explanation:

Please Help, This is Due in 4 Hours.
At the deli counter, Dalton bought 0.38 of a pound of roast beef and 0.7 of a pound of ham. Compare the amounts of roast beef and ham that Dalton bought.

Record the comparison using the <, >, or = symbol and explain how you know.

Answers

Answer:

0.38 < 0.70

Step-by-step explanation:

you have to estimate​

Answers

The answer is 75388.
802x94
=75388

Answer:

75,388

Step-by-step explanation:

Long Multiplication form

[tex]\mathrm{Mutliply\:the\:bold\:numbers}[/tex]

[tex]\frac{\begin{matrix}\space\space&8&0&\textbf{2}\\ \times \:&\space\space&9&\textbf{4}\end{matrix}}{\begin{matrix}\space\space&\space\space&\space\space&8\end{matrix}}[/tex]

[tex]\frac{\begin{matrix}\space\space&8&\textbf{0}&2\\ \times \:&\space\space&9&\textbf{4}\end{matrix}}{\begin{matrix}\space\space&\space\space&0&8\end{matrix}}[/tex]

[tex]\mathrm{Multiply}\\\\8\cdot \:4=32[/tex]

[tex]\mathrm{Carry\:}3\mathrm{\:to\:the\:column\:on\:the\:left\:and\:write\:}2\mathrm{\:in\:the\:result\:line}[/tex]

[tex]\frac{\begin{matrix}\space\space&3&\space\space&\space\space&\space\space\\ \space\space&\space\space&\textbf{8}&0&2\\ \times \:&\space\space&\space\space&9&\textbf{4}\end{matrix}}{\begin{matrix}\space\space&\space\space&2&0&8\end{matrix}}[/tex]

[tex]\mathrm{Add\:the\:carried\:digit,\:}3\mathrm{,\:to\:the\:result}[/tex]

[tex]\frac{\begin{matrix}\space\space&3&\space\space&\space\space&\space\space\\ \space\space&\space\space&8&0&2\\ \times \:&\space\space&\space\space&9&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\end{matrix}}[/tex]

[tex]\mathrm{Multiply\:the\:top\:number\:by\:the\:bolded\:digit\:of\:the\:bottom\:number}[/tex]

[tex]\frac{\begin{matrix}\space\space&\space\space&\textbf{8}&\textbf{0}&\textbf{2}\\ \space\space&\times \:&\space\space&\textbf{9}&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\end{matrix}}[/tex]

[tex]\mathrm{Mutliply\:the\:bold\:numbers}:\quad \:2\cdot \:9=18[/tex]

[tex]\mathrm{Carry\:}1\mathrm{\:to\:the\:column\:on\:the\:left\:and\:write\:}8\mathrm{\:in\:the\:result\:line}[/tex]

[tex]\frac{\begin{matrix}\space\space&\space\space&\space\space&1&\space\space\\ \space\space&\space\space&8&0&\textbf{2}\\ \space\space&\times \:&\space\space&\textbf{9}&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\\ \space\space&\space\space&\space\space&8&\space\space\end{matrix}}[/tex]

[tex]\mathrm{Add\:the\:carried\:number\:to\:the\:multiplication}:\quad \:1+0\cdot \:9=1[/tex]

[tex]\frac{\begin{matrix}\space\space&\space\space&\space\space&\textbf{1}&\space\space\\ \space\space&\space\space&8&\textbf{0}&2\\ \space\space&\times \:&\space\space&\textbf{9}&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\\ \space\space&\space\space&1&8&\space\space\end{matrix}}[/tex]

[tex]\mathrm{Mutliply\:the\:bold\:numbers}:\quad \:8\cdot \:9=72[/tex]

[tex]\mathrm{Carry\:}7\mathrm{\:to\:the\:column\:on\:the\:left\:and\:write\:}2\mathrm{\:in\:the\:result\:line}[/tex]

[tex]\frac{\begin{matrix}\space\space&7&\space\space&1&\space\space\\ \space\space&\space\space&\textbf{8}&0&2\\ \times \:&\space\space&\space\space&\textbf{9}&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\\ \space\space&2&1&8&\space\space\end{matrix}}[/tex]

[tex]\mathrm{Add\:the\:carried\:digit,\:}7\mathrm{,\:to\:the\:result}[/tex]

[tex]\frac{\begin{matrix}\space\space&7&\space\space&1&\space\space\\ \space\space&\space\space&8&0&2\\ \times \:&\space\space&\space\space&9&4\end{matrix}}{\begin{matrix}\space\space&3&2&0&8\\ 7&2&1&8&\space\space\end{matrix}}[/tex]

[tex]\mathrm{Add\:the\:rows\:to\:get\:the\:answer.\:\:\:Fill\:in\:trailing\:zeros\:\:(not\:required)}[/tex]

[tex]\frac{\begin{matrix}\space\space&\space\space&8&0&2\\ \space\space&\times \:&\space\space&9&4\end{matrix}}{\begin{matrix}0&3&2&0&8\\ 7&2&1&8&0\end{matrix}}[/tex]     [tex]=75,388[/tex]

See Image for the addition:

What is the value of w?

Answers

Answer:

w= -2. Solve for w: (w-5) (w+2) = w^2-4. Expand out terms of the left hand side: w^2-3 w-10 = w^2-4. Subtract w^2-4 from both sides

Step-by-step explanation:

plz help its so confusing

Answers

Answer:

20, 4

Step-by-step explanation:

Let's assume total number of students registered is n

Last week n/4 people are absent

so people present are n - n/4 = 3n/4

Today after return of Jen  1/5 n are absent

so people present are n - n/5 = 4n/5

which is also equal to last week present people + 1 (Jen)

= 3n/4 + 1

= (3n + 4) / 4

so (3n + 4) / 4 = 4n/5

=> 3n + 4 = 4n/5 * 4

=> (3n + 4) * 5 = 16n

=> 15n + 20 = 16n

=> 20 = n

=> n = 20

So total students registered for class is 20

today no of people absent = n/5 = 20/5 = 4

8.5x-y= 16
O -y= 5x + 16
O x=5y-16
O y= 5x +16
O y= 5x- 16

Answers

Answer:

y = 5x - 16

Step-by-step explanation:

1. To solve this question, we must convert 5x - y = 16 into slope-intercept form (y = mx + b).

2. (Solving for y)

Step 1: Subtract 5x from both sides.

[tex]5x - y - 5x = 16 - 5x[/tex] [tex]-y = -5x +16[/tex]  

Step 2: Divide both sides by -1.

[tex]\frac{-y}{-1} =\frac{-5x+16}{-1}[/tex] [tex]y = 5x - 16[/tex]

Therefore, the answer is y = 5x - 16.

A furniture store receive
order for 456 chairs. They
can fit 9 chairs in a large
shipping box. How many
shipping boxes will they
need to ship all of the
chairs?

Answers

The answer is 50 or 51 because it’s 50.666 so you could round it to the nearest 10.

A study found that 85% of sports business executives surveyed believed that sponsor development of ________ in new venues will be a benefit of top-tier sponsorships in coming years.

Answers

sports. it makes sense

Which set of ordered pairs lies along the same vertical line?

Answers

Answer:

All ordered pairs with the same x value

Step-by-step explanation:

A vertical line can be represented by the equation x=k for some value k. So any pair with x value equal to k satisfies the equation and is thus in the set

The table shows a proportional relationship between x and y what is the unit rate of y per x ( 4, 5 , 6 , 7 ) (8 , 10 , 12 , 14 ) ?

Answers

Answer:

6

Step-by-step explanation:

Problem 1
Each scholar in fifth grade participates in one of the three extracurricular activities: sports,
chess, or art.
2/5 of the fifth grade scholars participate in sports
1/4 of the fifth grade scholars participate in chess
What fraction of the fifth grade scholars participate in art?

Answers

Answer:

7/20 participate in art.

Answer:

7/20 hcjauwauhfehhfieafuewaueiwavibiurbviubwibvebaiubj

To mail a letter to Vienna, Austria the post office charges a flat rate of $8.00 and an additional $0.50 for every ounce the letter weighs. The cost of mailing a letter is determined by the equation y=0.50x+8.00, where y is the cost of the postage and x is the weight of the letter in ounces.

Step 2 of 2 : Use the graph to estimate to the nearest quarter of a dollar ($0.25) the cost to send a letter that weighs 3 ounces.

Answers

Answer:

It will cost 38 quarters

Step-by-step explanation:

y = 0.50x + 8.00

y = 0.5(3) + 8.00

y = 1.5 + 8.00

y = $9.50

$9.50/.25

38 quarters

Hope this is the answer you are looking for.

Pls help this is timed and I don’t understand

Answers

Answer:

178

Step-by-step explanation:

All right!!

Line l is parellel to line m.

So if r is 91 then we can infer that s helps it make a 180 degree angle/straight line.

So 180 - 91 = 89

s = 89

If the two lines are parallel then s = u because of inner angles/outer angles stuff.

If s=u then u = 89 as well.

And because of vertical angles, t =u so t = 89.

Now the question asks to find t + u, so t + u = 89 + 89 = 178

Hope that helps!!

100 Points
Emily wants to buy turquoise stones on her trip to New Mexico to give to at least 4 of her friends. The gift shop sells small stones for $4 and large stones for $6, and Emily has no more than $30 to spend.
Write and graph a system of Linear Equations and write two possible solutions.

Answers

Answer:

Step-by-step explanation:

what is she going to buy? Small stones or big stones? if its small stones then she can buy 7 small stones. If its big stones then she can buy 5 big stones.

Evaluate.

12⋅(1/4+1/3)power 2 +23 help ASAP

Answers

Answer:

325/12

Step-by-step explanation:

12(1/4 + 1/3)² + 23

12(7/12)² + 23

12(49/144) + 23

49/12 + 23

325/12 = 27 1/12 = 27.08333333

WHAT DOES EXPONENT C MEAN LETS SAY A NUMBER IS IN PARANTHESIS AND WITHIN THAT PARENTHESIS THERES A C AS AN EXPONENT WHAT DOES THAT MEAN

Answers

Answer:

It means that the number is raised to the power of c. For example, if it is (2^3), the answer would be 8.

Step-by-step explanation:

sara buys two bags of chips and three boxes of pretzels for $ 85. He then buys another bag of chips and two more boxes of pretzels for $ 50. Find the cost of each bag of chips and each box of pretzels.

Answers

Answer:

Chips= $20    Pretzel=$15

Step-by-step explanation:

Sara bought 2 Chips + 3 Pretzels = $85

(Extra) 1 Chips + 2 Pretzels = $50

Pretzels are twice as much as chips so above ^^ Can be expressed

as Pretzel (15) + Pretzel (15) + Chips(20) = $50

Now lets go back to the first line Putting in the new prices

Chips(20) + Chips(20) + Pretzel(15) + Pretzel(15) = Pretzel(15) = $85


You buy a $266,000 home.with. a.down payment of 21%. Find the amount of the down payment and the mortgage amount.
down payment mortgage amount

Answers

The answer is $210,140

If 3 bananas cost of 50 cents, how many bananas can be bought for 20 dollars?

Answers

Answer:

120 bananas

Step-by-step explanation:

$20 = 2000 cents

Create a proportion:  

Cross multiply, then divide: 3 × 2000 = 6000, 6000 ÷ 50 = 120

So, you can buy 120 bananas for $20

hope this helps

1
4
3
Time (h)
2
.
3
Cars
Washed
6
9
12
a. Proportional?
b. Equation:
C. Number of hours:
3. Cars washed:

Answers

Step-by-step explanation:

Proportional = time/cars washed

= 1/3

no. of hours= 90hours

cars washed = 30 cars washed

plz help asap plzz plllzzzzz

Answers

67/20 then 3 2/5 then 3.463 then 3.49

What is the period of the trigonometric function f(t) = 3sin()?
A) 2 pi
B) 3 pi
C) 4 pi
D) pi / 2

Answers

Your answer would be B) 3 pi

Crafts Unlimited sells ribbon by the foot. Serena needs ribbon

for an art project. The cost of the ribbon is 5 feet for $3.00.

Feet of Ribbon 1 5 10 15

Cost (in dollars) $3.00

a. Complete this ratio chart based on the information provided.

b. How many feet of ribbon can Serena buy for $12.00?

c. If Serena needs 17 feet of ribbon, how much will she pay?

How do you know?

Answers

The completed ratio chart is provided below:

Feet of Ribbon    1             5   10    15

Cost                     $0.60   $3  $6   $9

The amount Serena would get if she has $12 is 20 yards.

The amount Serena would pay if she wants 17 feet of the ribbon is $10.20.

The first step is to determine the unit cost of a ribbon.

Unit cost of a ribbon = $3 / 5 = $0.60

The cost of 10 feet of the ribbon = $0.60 x 10 = $6

The cost of 15 feet of the ribbon = $0.60 x 15 = $9

The feet of ribbon Serena can buy if she has $12 = $12 / $0.60 = 20 yards.

The cost of 17 feet of the ribbon = $0.60 x 17 = $10.20

To learn more, please check: https://brainly.com/question/12528427

A rectangular photograph has an area of 48 square inches. If the width of the photograph is 8 inches less than its height, find the dimensions of the photograph
The height of the photograph is

Answers

Answer:

H= 12, W=4

Step-by-step explanation:

let H = height

let H-8 = width

We know that the area of the rectangle is A = H*W

now we substitute the statements we have made:

H(H-8) = 48

H^2 - 8H - 48 = 0

Simplify:

(H-12)(H+4) = 0

So now we have 2 options H = -4 or H = 12, having a -ve number as height does not make sense so we use 12.

We know that width is H-8 so now we do 12-8 which is 4.

Therefore width is 4 and height is 12.

A satellite orbits the Earth with an elliptical orbit modeled by x squared over 47 million three hundred thirty four thousand four hundred plus y squared over forty three million nine hundred fifty six thousand nine hundred equals 1 comma where the distances are measured in km. The Earth shares the same center as the orbit. If the radius of the Earth is 6,370 km, what is the maximum distance between the satellite and the Earth?

6,880 km
6,630 km
510 km
260 km

Answers

An elliptical of the satellite in the field of astrodynamics is known as a

Kepler orbit.

The distance between the satellite and the Earth is; 510 km.

Reasons:

The function that represents the elliptical orbit of the satellite is presented

as follows;

[tex]\dfrac{x^2}{47,334,400} + \dfrac{y^2}{43,956,900} = 1[/tex]

The radius of the Earth, R = 6,370 km.

The equation of an ellipse is presented as follows;

[tex]\mathbf{\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2}} = 1[/tex]

Where, in a horizontal ellipse;

a = The major axis (the longer axis), which gives the maximum distance

between the satellite and Earth.

b = The minor axis (the shorter axis)

By comparison, we get;

a² = 47,334,400 km²

Therefore;

a = √(47,334,400 km²) = 6,880 km.

The distance between the satellite and the surface of the Earth, d, is given

as follows;

d = a - R

Which gives;

d = 6,880 km - 6,370 km = 510 km

Distance between the satellite and the surface of the Earth, d = 510 km.

Learn more here:

https://brainly.com/question/13999216

Answer:510 KM

Step-by-step explanation:

The values of x and y which satisfy the equations
x + 2 y = 27 and 2x - y = 19 are respectively
(A) 15 and 10
(B) 10 and 15
(C) 7 and 13
(D) 13 and 7

Answers

D) 13 and 7

This is because when the value of x is 13 and the value of y is 7, these equations both become true.

Lol pls hurry!!
I need this asap

Answers

Answer: 2, D

Step-by-step explanation: f=-2, so 4x-2 would give -8 add 10 would give 2 as it is for the negative

Incredibly, Enzo wins 5 tickets from every game, and Beatriz wins 11 tickets from every game. When they stopped playing games, Enzo and Beatriz had won the same number of total tickets.
What is the minimum number of games that Enzo could have played?

Answers

Answer:

Enzo must have played 11 games.

Step-by-step explanation:

Since Enzo earns 5 tickets from every game and Beatriz wins 11 tickets from each game, we have to find the LCM (Least Common Multiple) since they earned the same amount of tickets when they stopped.

The LCM of 11 and 5 is 55.

To see how many games they both played we have to divide the amount of tickets they earn each time from the LCM (55).

Since Enzo earns 5 tickets from each game and both Enzo and Beatrice earned 55, Enzo must've played 11 games.

(If that makes sense lol)

what is the equation of this line in slope-intercept form

Answers

Answer:

y=3/5x+3 i think

Step-by-step explanation:

3 is the y intercept, and to find the slope its rise over run so 3 up and 5 to the side, i hope this helps!!

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