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

A right triangle with a hypotenuse of square root of 61 has an area of 15 square inches. Find the lengths of the other two sides.



Answers

The lengths of the other two sides are 6 inches and 5 inches

Let b represent the base of the right triangle, h represent the height of the right triangle and x represent the hypotenuse.

Since the area of the triangle is 15 in², hence:

Area(A) = (1/2) * base * height

15 = (1/2)bh

bh = 30  

b = 30/h         (1)

Applying Pythagoras theorem also:

x² = b² + h²

(√61)² = b² + h²

61 = (30/h)² + h²

61 = 900/h² + h²

61h² = 900 + h⁴

h⁴ - 61h² + 900 = 0

h = -6, -5, 5, 6

Since the length cant be negative, hence h = 5 or h = 6

When h = 5; b = 30/5 = 6

When h = 6; b = 30/6 = 5

Therefore the lengths of the other two sides are 6 inches and 5 inches

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If f(x)=8-5x, what is f(4p+3)

Answers

Answer:

-20p - 7

Step-by-step explanation:

f(x)=8-5x

f(4p+3) = 8 - 5(4p + 3)

= 8 - 20p - 15

= -20p -7

Below is an excerpt from the home loan contract Kyle signed with his bank.
“The sum of the principal of the loan and all applicable interest will be divided into 360 monthly payments according to the amortization schedule.
The Buyer shall make each monthly payment by the first day of the month it is due. A payment not received by the first day of the month it is due will be considered delinquent. A payment not received by the fifteenth day of the month it is due will be subject to a late charge of $75.00. If a payment is not received by the last day of the month it is due, the Buyer and transaction shall be reported to the appropriate credit agencies.
In the event of delinquency, the Buyer may divide any payment not received into not more than three equal installments to be included on the next consecutive monthly payments. Notification of the Buyer’s intent to do so must be made by written letter or phone call to the phone number and address listed below within one business week of delinquency.”
Kyle failed to make his payment by the first of this month. Kyle’s monthly payment is currently $1,200. If Kyle decides to split the missing payment into two parts and pay it over the next two months, which of the following statements is false?
a.
Kyle will be reported to a credit agency.
b.
Kyle will be charged a $75 late fee for the missing payment.
c.
Kyle has one business week to inform his bank of the late payment.
d.
Kyle’s monthly payments for the next two months will remain $1,200.

Answers

The answer is A hope that helps

Gus, Verid and their three children went to see a show.,
The children were charged half-price. If the total cost of the tickets was $94.50, how
much did each adult ticket cost?

Answers

Answer:

28.35 each adult

Step-by-step explanation:

About $27 for the adults tickets I believe

Dеtеrmіnе tһе rаnցе օf tһе fսnсtіօn f(х) = 3х - 5 ԝһеn tһе ԁօmаіn іѕ {-4, 2, 9}.

Answers

Answer:

range is { - 17, 1, 22 }

Step-by-step explanation:

Substitute the values from the domain (input) into f(x) for range

f(- 4) = 3(- 4) - 5 = - 12 - 5 = - 17

f(2) = 3(2) - 5 = 6 - 5 = 1

f(9) = 3(9) - 5 = 27 - 5 = 22

range is { - 17, 1, 22 }


Find the value of x and y

Answers

X= 45 degrees y= 85 degrees

The fact that the sampling distribution of the sample mean can be approximated by a normal probability distribution whenever the sample size is large is based on the _____.

Answers

Answer:

can you add a pic

Step-by-step explanation:

What is the algebraic expression for the following word phrase: the product of 7 and the difference of x and y?

Answers

Answer:

7(x-y)

Step-by-step explanation:

difference of x and y=x-y

product of 7=7 times the expression

so,

Answer is 7(x-y)

whats the answer can someone answer please​

Answers

Answer:

b and c

Step-by-step explanation:

Which methods of evaluating a capital investment project ignore the time value of money?

Answers

Answer: Varies

Step-by-step explanation:

It involves any that is not concerned with making a profit.

what are the solutions to the system

Answers

[tex]10+y=5x+x^2~~~~.....(i)\\\\5x+y =1~~~....(ii)\\\\\text{From (i):}\\\\10+y = 5x +x ^2\\\\\\\implies y = 5x +x ^2 -10\\\\\text{Substitute y in equation (ii):}\\\\5x+5x+x^2-10=1\\\\\implies x^2 + 10x -11 =0\\\\\implies x^2 +11x -x -11=0\\\\\implies x(x+11) -(x+11) =0\\\\\implies (x-1)(x+11) =0\\\\\implies x =1 ~~ \text{or}~~ x = -11\\\\\text{Substitute x =~1 in equation (ii):}\\ \\5+y =1 \implies y =-4[/tex]

[tex]\\\text{Substitute}~ x =-11~ \text{in equation (ii):}\\\\\ 5(-11) + y = 1 \implies y = 1 +55 = 56 \\\\\text{Hence} (x,y) = (-11,56) ~~\text{and}~~ (x,y)=(1,-4)[/tex]

The graph below shows a distance versus time graph. What is the average speed
between 2 p.m. and 4 p.m.?

Answer Choices and Graph in the picture.

Answers

Answer:

option B is correct answer.

Step-by-step explanation:

(200-75)/2 = 62.5 miles per hour.

simplify 5 − 2(2.5 − 6.7)^2

Answers

Answer:

-30.28

Step-by-step explanation:

evans uncle bought 2 1/4 pounds of hamburger meat and used 1/3 of it to make lunch. How much hamburger meat did evans uncle use to make lunch?

Answers

Answer/Step-by-step explanation:

To find the answer subtract 2 1/4 by 1/3

number of heads: 0,1,2,3 Frequency: 16,38,34,12 Find the median​

Answers

Answer:

the median of 0 1 2 3 16 38 34 12 is 9.5

87 is 12.5% of what number?

Answers

Answer:

10.88

Step-by-step explanation:

A woman can bicycle 78 miles in the same time as it takes her to walk 24 miles. She can ride 9 mph faster
than she can walk. How fast can she walk?
mph

Answers

Answer:

4mph

Step-by-step explanation:

Magnesium has a charge of +2, and chlorine has a charge of –1. To form a chemical bond, how many magnesium ion(s) would the compound have?

A) 0
B) 1
C) 2
D) 3

Answers

In a chemical bond, the charges are balanced - neutral. This means that if we have a charge of +2 we will need a charge of -2 to cancel it out.

This means we need 1 magnesium ion and 2 chlorine ions.

Answer: B) 1

Hope this helps!

A. What does each of these points represent in the is situation: (0, 0), (1, 55), and (5, 275)?

What does each of these points represent in this situation (0, 0), (1, 55) and (5, 275)? Graph shows Mr. Brown’s road trip, with time (h) on the x-axis and distance (mi) on the y-axis. Points marked on the line graph are at (0, 0), (1, 55) and (5, 275).

Answers

Step-by-step explanation:

[tex]︎{\blue{\rule{25pt}{1pt}{\purple{\rule{30pt}{3pt}{\red{\rule{50pt}{9999pt}{\purple{\rule{30pt}{3pt}{\blue{\rule{25pt}{1pt} }}}}}}}}}}︎︎︎[/tex]

Select the answer that describes the properties of a parallelogram.

Answers

Answer:

A is the correct answer because all sides are congurent

A parallelogram has two pairs of parallel sides.

opposite sides and opposite angles are congruent.

Option D is the correct answer.

We have,

Parallelograms have several notable properties:

Opposite sides are parallel:

In a parallelogram, opposite sides are parallel, meaning they never intersect and have the same slope.

Opposite sides are congruent:

The opposite sides of a parallelogram have equal lengths. This property applies to both pairs of opposite sides.

Opposite angles are congruent:

The opposite angles of a parallelogram have equal measures. This property applies to both pairs of opposite angles.

- Consecutive angles are supplementary.

- Diagonals bisect each other.

- Diagonals create congruent opposite triangles.

- Diagonals create equal opposite angles.

Thus,

A parallelogram has two pairs of parallel sides.

opposite sides and opposite angles are congruent.

Learn more about parallelograms here:

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4. 8 is what percent of 18?

Answers

Answer:

the answer is 26.6666667 and when rounded off is 26.7

Guided Practice


Write the equation of the parabola in vertex form. The vertex is (3, 2) and the parabola passes through the point (0, 38). Create a rough sketch of the graph to use as a visual aid.


A.
y = 4(x + 3)2 + 2


B.
y = (x + 3)2 + 2


C.
y = 19(x − 2)2 + 3


D.
y = 4(x − 3)2 + 2

Answers

Answer:

D.

y = 4(x − 3)^2 + 2

Step-by-step explanation:

A.    y = 4(x + 3)^2 + 2   = ( − 3 ,  2 )

B.    y = (x + 3)^2 + 2     = ( − 3 ,  2 )

C.    y = 19(x − 2)^2 + 3  = ( 2 ,  3 )

D.    y = 4(x − 3)^2 + 2   = ( 3 ,  2 )

Write the quadratic function in f(x)=a(x−h)2+k form whose graph is shown below.



The graph shows a downward facing parabola with vertex (negative 3, 4) and y-intercept (0, negative 5).

Answers

[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{o pens~\cap}\qquad \stackrel{"a"~is~positive}{o pens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex](\stackrel{h}{-3}~~,~~\stackrel{k}{4})\qquad \implies f(x) = a[x-(-3)]^2+4\implies f(x) = a(x+3)^2+4 \\\\\\ \textit{we know that} \begin{cases} x=0\\ f(x) = -5 \end{cases}\implies -5=a(0+3)^2+4\implies -5=9a+4 \\\\\\ -9=9a\implies \cfrac{-9}{9}=a\implies \boxed{-1=a}~\hfill \blacktriangleright f(x)=-(x+3)^2+4 \blacktriangleleft[/tex]

helpppp please!!!!!!!

Answers

Answer:

a

Step-by-step explanation:

a

QUICKLY!!!! A paper airplane was thrown from the top of a tall building.The height of the paper airplane above the ground can be found using the function `y=-1.5x+60` , where `x` is the time in seconds the airplane has been in the air.
Jayden made this table to figure out how high the plane was at different times.
0 | 60
10 | 45
20 | 30
30 | 15
40 | 0


How would the equation `y=-1.5x+60` change if the height at 0 seconds was 75?

Answers

Answer:

y = -1.5x + 75

Step-by-step explanation:

Well, the rate at which the plane falls doesn't change, so anything attached to the time variable remains constant.

In this case, the first part of the function (-1.5x) remains the same.

Since the height of the tall building was 60 and the value in the equation, we can see that the last value corresponds to the number of meters above ground.

Since the height has changed to 75, the last part of the equation (60) changes to 75, and the rest remains constant.

That gives us the following equation: y = -1.5x + 75

What is the inequality shown?

Answers

-4<x>/=5

○ means either< or > but since the line is moving to the right it means that x is greater than -4 (<x)● means greater than or equal to since the line stoped at 5 it means x can be equal to 5 or less then 5

An arrow is shot at an angle 32 degree from the horizontal. If it's height is increasing at 200 km/hr, how fast is the arrow itself going?

Answers

Check the picture below.

Bearing in mind that the angle is not changing all the wihle, namely is constant

[tex]tan(32^o)=\cfrac{y}{x}\implies x = \cfrac{y}{tan(32^o)}\implies x = \cfrac{1}{tan(32^o)}y \\\\\\ \stackrel{\textit{taking the derivative on bot sides}}{\cfrac{dx}{dt}=\cfrac{1}{tan(32^o)}\cfrac{dy}{dt}}\implies \cfrac{dx}{dt}=\cfrac{1}{tan(32^o)}200\implies \cfrac{dx}{dt}\approx 320.07~\frac{km}{hr}[/tex]

BEST ANSWER GETS BRAINLIEST!!!!!!!

ruben the trainer has two solo workout plans that he offers his clients: Plan A and Plan B. Each client does either one or the other (not both). On Monday there were 9 clients who did Plan A and 7 who did Plan B. On Tuesday there were 3 clients who did Plan A and 5 who did Plan B. Reuben trained his Monday clients for a total of 12 hours and his Tuesday clients for a total of 6 hours. How long does each of the workout plans last?

Answers

Let plan A be x

Plan B be y

Equations are

[tex]\\ \sf\longmapsto 9x+7y=12--(1)[/tex]

[tex]\\ \sf\longmapsto 3x+5y=6--(2)[/tex]

Multiplying (1) by 1 and (2) by 3

9x+7y=129x+15y=18

Subtracting

[tex]\\ \sf\longmapsto -8y=-6[/tex]

[tex]\\ \sf\longmapsto y=3/4[/tex]

Putting in eq(2)

[tex]\\ \sf\longmapsto 3x+5(3/4)=5[/tex]

[tex]\\ \sf\longmapsto 3x+15/4=5[/tex]

[tex]\\ \sf\longmapsto 3x=5-4/15[/tex]

[tex]\\ \sf\longmapsto 3x=71/15[/tex]

[tex]\\ \sf\longmapsto x=71/45[/tex]

4. Find a value of x that is a solution to 5(x - 3) = x + 13. Describe how you found the solution​

Answers

Answer:

x=7

Step-by-step explanation:

5(x - 3) = x + 13

Distribute the 5

5x-15 = x+13

Subtract x from each side

5x-15-x = x+13-x

4x-15 =13

Add 15 to each side

4x-15+15 = 13+15

4x= 28

Divide each side by 4

4x/4 = 28/4

x = 7

Answer:

x = 7

Step-by-step explanation:

First

5(x - 3) = 5x - 15 by distributing the 5 on each variable and number in the parenthesis.

Then

5x - 15 = x + 13 which you add 15 (or 5x - 15 + 15 = x + 13 +15) on both sides to have the numbers on one side because it must be the opposite to cancel out the 15 on the original/previous side.

Afterwards

5x = x + 28 now we must get the x to the other side and using the same logic as before you move the x to the other side by subtracting it (or 5x - x = x + 28 - x) to the other side equaling 4x = 28.

Finally

4x = 28 you need to divide each side by 4 to cancel out the 4 on the x since we must find out what x is (or 4x/4 = 28/4), and you should get x = 7.

Andrew made a list of chores he needed to complete before his father comes home from work. On the chart, he showed the time it would take him to complete each chore.

If Andrew’s father is expected home at 6:00 p.m., what is the latest time Andrew could begin his chores?



Use the Hint to view the conversion chart.

Answers

Answer: the answer is 3:20pm!

Step-by-step explanation:

i got the question correct just now!

Answer:

3;20

Step-by-step explanation:

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