Define the function
3
g(x) = x^3+ x
. If
f(x) = g^-1(x)
and
f(2) = 1
what is the value of
f'(2)
?

Answers

Answer 1

Given that

[tex]g(x) = x^3 + x[/tex]

the inverse [tex]g^{-1}(x)[/tex] is such that

[tex]g\left(g^{-1}(x)\right) = g^{-1}(x)^3 + g^{-1}(x) = x[/tex]

or

[tex]g\left(f(x)\right) = f(x)^3 + f(x) = x[/tex]

Differentiating both sides using the chain rule gives

[tex]3f(x)^2f'(x) + f'(x) = 1 \\\\ f'(x) \left(3f(x)^2+1\right) = 1 \\\\ f'(x) = \dfrac1{3f(x)^2+1}[/tex]

Then the derivative of f at 2 is

[tex]f'(2) = \dfrac1{3f(2)^2+1} = \boxed{\dfrac14}[/tex]


Related Questions

Graph the equation.
y=-3x - 4

Answers

Step-by-step explanation:

Goes down 4 since it's a negative

At the art store, Aisha bought a few blocks of clay to make a teapot and matching mugs for her aunt. She used 2 1/2 pounds of clay to make the mugs and 1 1/2 times as much clay for the teapot. If each block of clay weighed 2 pounds, how many blocks did Aisha use in all?

Answers

Answer:

3 1/8

Step-by-step explanation:

Pls give brainliest

Add 2/5 + 3/6 + 1/4 0 1 11/60 O 13/20 o 17/20 1 13/60 ​

Answers

Answer:

Step-by-step explanation:

-(8/11)/(-3/5) = -(8/11)*(-5/3)

-(8/11)*(-5/3) = 40/33  [ or 1 7/33]

I can't interpret the rest of the question.  I see:

"Add 2/5 + 3/6 + 1/4 0 1 11/60 O 13/20 o 17/20 1 13/60"

A coin weighs 4/15 ounces. A different coin weighs 1/15 ounces. How much do the two coins weigh together? Write your answer as a fraction in simplest form.

Answers

Answer:

1/3

Step-by-step explanation:

4/15 + 1/15 = 5/15

simplified its 1/3

4[3(5+2)]=95-4x3 is this statement true or false

Answers

Answer:

TRUEeeee

Step-by-step explanation:

bec. there beetwen have a

Write the equation of the line that passes through the given points.
(-6,0) and (0,3)

Answers

Answer:

y = 1/2x + 3

Step-by-step explanation:

First we find the slope:

(y2 - y1)/(x2 - x1) =

3-0 / 0 + 6

3/6 = 1/2

then the y intercept is 0,3 so we add 3

y = 1/2x +3

Difination of occupation​

Answers

Answer:

Occupation -> a job or profession: his prime occupation was as editor.

• a way of spending time: a game of cards is a pretty harmless occupation.

2 the action, state, or period of occupying or being occupied by military force: the Roman occupation of Britain | the Nazi occupation.

• the action of entering and taking control of a building: the workers remained in occupation until October 16.

3 the action or fact of living in or using a building or other place: a property suitable for occupation by older people.

Step-by-step explanation:

hope it helps

Answer:

It is a job or profession

Step-by-step explanation:

If you have an occupation then that means you have a job.

Hope this helps!

PLEASE HELP ME ITS ALGEBRA 2 THANK YOUUUU

Answers

Answer:

10

Step-by-step explanation:

Proprotions:

81/9 = 90/c(?)

so

81c= 9(90)

81c=810

810/81=

10

so c=10

Hope it helped!!

14+3(12+2)-5 to the power of 2=
show your work.

Answers

pemdas

p- 12+2=14
14+3(14)-5^2
e-5^2= 25
m-3x14=42
a and s going left to right
14+42-25
14+42=56
56-25= 31
your answer is 31

Please help me I don’t understand this

Answers

Answer:

V/pi^2 = h

Here is your answer my dear friend-

C. r^2 = Vpi h

Find all exact solutions on the interval [0,2π) csc^2(x)-9= -5

Answers

Answer:

d

Step-by-step explanation:

The population of a town has been decreasing By 5% each year for the last two years the current population of the town is 1510 people what was the population two years ago

Answers

First start with your starting value which is 1510 and then get the multiplier.. to get the multiplier for a precent decrease you have to subtract it from 100%.. once you’ve got the multiplier you put it into the equation and because it’s asking for 2 years ago you make the exponent -2

If you earn 12000 per month and you except your earnings to increase by 10% per year,how much do you think you will be making per year in 5 years?(round hundredths place)

Answers

Step-by-step explanation:

1 month=12000/=

12months=144000000/=

1 year=144000000

5years=720000000/=

7200000.000/=

A piece of wire 3/7 m long as we cut into six pieces of the same length what is the length of each piece

Answers

Answer:

Each piece is 1/7 m long

Step-by-step explanation:

Wire length = 3/7 m

Pieces = 6

1 piece = 3/7 ÷ 6

= 3/7 ÷ 6/1

= 3/7 × 1/6

= 1/7 m

A line has a slope of -1/7. It passes
through the points (1,2) and (3,y).
What is the value of y?

Answers

Answer:

y = 12/7

Step-by-step explanation:

Slope = (change in Y) / (change in x)

-1/7 = (y - 2) / (3 - 1)

-1/7 = (y - 2)/2

Cross multiply:  -1(2) = 7(y - 2)

                             -2 = 7y -14

                        -2 + 14 = 7y

                                12  = 7y

                               12/7 = y

Let's find y

[tex]\\ \sf\longmapsto m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\\ \sf\longmapsto \dfrac{-1}{7}=\dfrac{y-2}{3-1}[/tex]

[tex]\\ \sf\longmapsto \dfrac{-1}{7}=\dfrac{y-2}{2}[/tex]

[tex]\\ \sf\longmapsto -2=7y-14[/tex]

[tex]\\ \sf\longmapsto 7y=12[/tex]

[tex]\\ \sf\longmapsto y=12/7[/tex]

Azizah got 1 for the slope of the line through points (1, 2) and
(4 - 1)

Explain to her the mistake she
made and how to find the
slope correctly.

Answers

Answer:

I hope it help you

Step-by-step explanation:

btw thanks for the points

[tex]▪▪▪▪▪▪▪▪▪▪▪▪▪  {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪[/tex]

let's find slope (m) :

[tex] \dfrac{y2 - y1}{x2 - x1} [/tex]

[tex] \dfrac{2 - ( - 1)}{1 - 4} [/tex]

[tex] \dfrac{3}{ - 3} [/tex]

[tex] - 1[/tex]

slope of the line is -1, so Azizah got it wrong.

Which equation is the inverse of 2(x - 2)2 = 8(7 + y)?
O -2(x - 2)2 = -8(7 + y)
O yx2-
o yo hx²-x-6
y=-2+ 28+4x
O y=2+28+4x

Answers

Answer:

-2(x-2)2=-8(7+y)

Step-by-step explanation:

Inverse means just -

Lets solve afterwards

[tex]\\ \sf\longmapsto -2(x-2)2=-8(7+y)[/tex]

[tex]\\ \sf\longmapsto (-2x+4)2=-56-y[/tex]

[tex]\\ \sf\longmapsto -4x+8=-56-y[/tex]

[tex]\\ \sf\longmapsto -4x+y=-64[/tex]

[tex]\\ \sf\longmapsto 4x-y=64[/tex]

Answer:

-2(x - 2)2 = -8(7 + y)

Step-by-step explanation:

2(x - 2)2 = 8(7 + y)

(2x - 4)2 = 56 + y

4x - 8 = 56 + y

4x - 8 - 56 = y

4x - 64 = y

If option A

-2(x - 2)2 = -8(7 + y)

(-2x + 4)2 = - 56 - y

- 4x + 8 = - 56 - y

- 4x + 8 + 56 = - y

- 4x + 64 = - y

4x - 64 = y

(RIGHT)✔

EVALUATE THE FUNCTION REQUESTED. WRITE YOUR ANSWER AS A FRACTION IN LOWEST TERMS
I WILL GIVE BRAINLIEST I NEED THIS ANSWER ASAPP!

Answers

Answer:

• From trigonometry ratios:

[tex]{ \boxed{ \rm{ \sin(A) = \frac{opposite}{hypotenuse} }}} \\ [/tex]

opposite → 28hypotenuse → 35

[tex]{ \rm{ \sin(A) = \frac{28}{35} }} \\ \\ { \boxed{ \boxed{ \rm{ \: \sin(A) = \frac{4}{5} \: }}}}[/tex]

One student starts with a 3-lb hand weight and increases the weight by 2 lb per
month. Another starts with a 5-lb hand weight and increases the weight by 1 lb
per month. For each student, write an equation that represents how much
weight y the student is lifting after x months.

Answers

Answer:

Student A: y=3+2x

Student B: y=5+1x

The area of a square region is 529 mi2. What is the perimeter of the​ region?

Answers

Answer:

92 miles

Step-by-step explanation:

The area of a square is Length x Width.  All sides of a square are equal in length so length = width.  This is the same as saying x = length and x = width so   X times X equals the area.  X times X is X².

This tells you that X² = 529 mi².   X would equal the square root of 529 which is 23.     23  x  23  = 529.

Each side of the square is 23 miles.  The perimeter is equal to side 1 plus side 2 plus side 3 plus side 4.  Again since it is a square all sides are equal.

This mean you can just take 4  x  23 miles.  The perimeter = 92 miles

Answer:

The perimeter of the region is 92 miles

Step-by-step explanation:

For a given Square

Area = 529 mi²

Now, To find area we use formula

A = (l)²

529 = l²

√529 = l

23 = l

l = 23 miles

Now, To find perimeter of square we use,

P = 4l

P = 4 (23)

P = 92 miles

Thus, The perimeter of the region is 92 miles

-TheUnknownScientist 72

Which of the following values of x would not be in the domain of the function defined by f(x) = x-2/x+3?

Answers

Answer:

x = 7 is not there in the domain of the function.

Step-by-step explanation:

The value of x = -3 would not be in the domain of the function

We have the following information available from the question is:

The function is :

[tex]f(x) =\frac{x-2}{x+3}[/tex]

Determine the values of x would not be in the domain of the function

Now, According to the question:

The function is :

[tex]f(x) =\frac{x-2}{x+3}[/tex]

Let the denominator equal to zero: x + 3 = 0

Rearrange unknown terms to the left side of the equation.

x + 3 = 0

x = -3

Hence, The value of x = -3 would not be in the domain of the function.

Learn more about Function at:

https://brainly.com/question/30721594

#SPJ3

Question in the picture

Answers

Answer:

hope this helps.......

solve pls brainliest

Answers

|5|-9=-4
|5-9|=4
I’m pretty sure this is correct I’ve checked like three times

What is 3763.2 ÷0.512 with the work that goes with it.

Answers

Answer:

7350

Explanation:

3763.2/0.512 can be determined by simply dividing.

considering the line negative x -6 y equals -6 what is the line perpendicular with a slope of a line perpendicular to this line what is the slope of the line parallel to this line​

Answers

9514 1404 393

Answer:

  perpendicular: undefined

  parallel: zero

Step-by-step explanation:

The line y = -6 is a horizontal line. Between any two points on the line, the difference in y-values is zero, and the difference in x-values is non-zero. The slope of this horizontal line is the ratio of these differences:

  slope = 0/(non-zero) = 0

Any parallel line will have the same slope. The slope of a parallel line is 0.

__

A line perpendicular to a horizontal line is a vertical line. It will have the equation x = constant. Between any two points on the line, the difference in y-values is non-zero, and the difference in x-values is zero. The slope of this vertical line is the ratio of these differences:

  slope = (non-zero)/0 = undefined

The slope of the perpendicular line is undefined.

I’ll give BRAINLIEST pls help

Answers

Answer: EQUATIONS OF HORIZONTAL AND VERTICAL LINES. In general, linear equation is a function whose graph is a straight line and it is typically written as. y = mx + b, where m is the slope and b is the y-intercept. An horizontal lines goes left and right and its equation is in the form of. y = b,

Step-by-step explanation:

What is the slope of a line perpendicular to the line whose equation is 3y + 2x = -5?

Answers

Answer:

I think the slope of the line is -2/3

What is the next number in the sequence 3,5,8,12,17?

Answers

Answer:

it would be 23

Step-by-step explanation:

5-3=2

5-8=3

12-8=4

17-12=5

23-17=6

Find the percent decrease. Round to the nearest percent.


From 138 pounds to 130 pounds


The percent decrease is %.

Answers

Answer: 6%

Step-by-step explanation:

so you find the normal decrease. which is 138-130, 8 pounds.

then you divide by the original weight, you get 5.979…% so then round it up and tada! 6%

Select the correct answer. Which of the following is equal to the expression below? A. B. C. D.

Answers

Answer:

b

Step-by-step explanation:

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