Answers
Answer:
y'' = 18x(x³ + 8)⁴(17x³ + 16)
General Formulas and Concepts:
Calculus
Derivative Notation dy/dx
Derivative of a constant is 0.
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Chain Rule: [tex]\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Product Rule: [tex]\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Step-by-step explanation:
Step 1: Define
y = (x³ + 8)⁶
Step 2: Find 1st Derivative (dy/dx)
Chain Rule [Basic Power]: y' = 6(x³ + 8)⁶⁻¹ · (3x³⁻¹ + 0)Simplify: y' = 6(x³ + 8)⁵ · 3x²Multiply: y' = 18x²(x³ + 8)⁵Step 3: Find 2nd Derivative (d²y/dx²)
Product Rule [Chain/Basic Power]: y'' = 2 · 18x²⁻¹ · (x³ + 8)⁵ + 18x² · 5(x³ + 8)⁵⁻¹ · (3x³⁻¹ + 0)Simplify: y'' = 36x(x³ + 8)⁵ + 90x²(x³ + 8)⁴ · 3x²Multiply: y'' = 36x(x³ + 8)⁵ + 270x⁴(x³ + 8)⁴Factor: y'' = 6x(x³ + 8)⁴[6(x³ + 8) + 45x³]Distribute: y'' = 6x(x³ + 8)⁴[6x³ + 48 + 45x³]Combine like terms: y'' = 6x(x³ + 8)⁴[51x³ + 48]Factor: y'' = 6x(x³ + 8)⁴ · 3(17x³ + 16)Multiply: y'' = 18x(x³ + 8)⁴(17x³ + 16)Answer:
[tex]\frac{d^{2}y}{dx^{2}} = 180x.(x^{3} +8)^{4}[/tex]
Step-by-step explanation:
This question requires use of the chain rule:
[tex]y = (x^{3} + 8)^{6}[/tex]
Let:
[tex]u = x^{3} + 8[/tex]
Then:
[tex]y = u^{6}[/tex]
And:
[tex]\frac{dy}{dx} = \frac{dy}{du}. \frac{du}{dx}\\\\\frac{d^{2}y}{dx^{2} } = \frac{d^{2}y}{du^{2}}. \frac{d^{2}u}{dx^{2}}\\\\\frac{dy}{du} = 6u^{5}\\\\\frac{d^{2}y}{du^{2}} = 30u^{4}\\\\\frac{du}{dx} = 3x^{2}\\\\\frac{d^{2}u}{dx^{2}} = 6x[/tex]
So:
[tex]\frac{dy}{dx} = 6u^{5}.3x^{2}\\\\\frac{dy}{dx} = 18u^{5}x^{2}\\\\[/tex]
And:
[tex]\frac{d^{2}y}{dx^{2}} = 30u^{4}.6x\\\\\frac{d^{2}y}{dx^{2}} = 180u^{4}x[/tex]
Finally, substitute u for its equivalent expression in x:
[tex]\frac{d^{2}y}{dx^{2}} = 180x.(x^{3} +8)^{4}[/tex]
Related Questions
Change to slope-intercept form: 3x + 12y = -60
Answers
Answer:
y=-1/4x-5
Step-by-step explanation:
Answer:
y = -1/4x - 5
Step-by-step explanation:
12y = -3x- 60
divide 12 on each side
y = - 1/4x -5
make me brainliest if answer is correct
Solve this equation. Enter your answer in the box. 75 – 3.5y – 4y = 4y + 6 y =
Answers
Answer:
y = 30/7
Step-by-step explanation:
Which statement represents the given algebraic expression?
\large -3n-17
a
Seventeen less than three times a number
b
Negative three minus seventeen
c
A number less than seventeen is negative three
d-Negative three of a number decreased by seventeen
Answers
Write each situation as a rate.
John travels 150 miles in 3 hours.
Cameron ate 5 apples in 2 days.
Answers
Answer:
See below.
Step-by-step explanation:
Distance traveled for John
150 miles / 3 hours = 50 miles/hour
Cameron apples
5 apples / 2 days = 2.5 apples/day
The rate is the per time value thus the rate at John travels is 50 miles/hour while Cameron ate apples is 2.5 apples/day.
What is the arithmetic operator?Arithmetic operators are four basic mathematical operations in which summation, subtraction, division, and multiplication involve.,
Division = divide any two numbers or variables called division.
For example 4/8
Multiplication = to multiply any two or more numbers or variables called multiplication.
As per the given question,
John travels 150 miles in 3 hours.
John travels per hour = 150 / 3 = 50 miles
Therefore,50 miles/hour
Cameron ate 5 apples in 2 days.
Cameron ate an apple per day = 5/2
Therefore,2.5 apples/day.
Hence "The rate is the per time value thus the rate at John travels is 50 miles/hour while Cameron ate apples is 2.5 apples/day".
To learn more about the arithmetic operators,
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What is an equation of the line that passes through the points (6, -6) and (-6, -8)
Answers
Answer:
An equation of the line that passes through the points (6, -6) and (-6, -8) is [tex]\mathbf{y+6= \frac{1}{6}(x-6)}[/tex]
Step-by-step explanation:
We need to write an equation of the line that passes through the points (6, -6) and (-6, -8)
We would use point slope formula: [tex]y-y_1=m(x-x_1)[/tex]
where [tex]x_1=6, y_1=-6 \ and \ m \ is \ slope[/tex]
Finding slope using formula: [tex]Slope=\frac{y_2-y_1}{x_2-x_1}\\[/tex]
We have [tex]x_1=6, y_1=-6, x_2=-6, y_2=-8[/tex]
Putting values and finding slope
[tex]Slope=\frac{y_2-y_1}{x_2-x_1}\\\\Slope=\frac{-8-(-6)}{-6-6}\\Slope=\frac{-8+6}{-12}\\Slope=\frac{-2}{-12}\\Slope=\frac{1}{6}[/tex]
So, slope = 1/6
The required equation having [tex]x_1=6, y_1=-6 \ and \ m =\frac{1}{6}[/tex]
[tex]y-(-6)=\frac{1}{6}(x-6)\\y+6= \frac{1}{6}(x-6)[/tex]
An equation of the line that passes through the points (6, -6) and (-6, -8) is [tex]\mathbf{y+6= \frac{1}{6}(x-6)}[/tex]
12. You roll one die. How many ways can you get a 3 or an odd number?
Answers
Mr. Lassie goes out to eat with his family. Their dinner costs $144. Mr. Lassie is going to leave a 25% tip because the service was excellent. What is the total amount of the dinner out?
Answers
Answer:
$180
Step-by-step explanation:
25% ->0.25
144(0.25)=36
36+144=180
Select the correct answer.
Which transformation of Alx) will produce the same graph as g(x)?
Rx) = 3x
g(x) = RX) + 3
O A. (x) = 2Rx)
OB.(x) = f(x - 1)
OC. (x) = fx + 1)
D. h(x) = f(x) + 1
Answers
The correct transformation of A (x) will produce the same graph as g(x) is,
⇒ f (x) = f (x + 1)
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The functions are,
⇒ R (x) = 3x
⇒ g (x) = R(x) + 3
Now, We get;
⇒ g (x) = R(x) + 3
⇒ g (x) = 3x + 3
⇒ g (x) = 3 (x + 1)
Thus, The correct transformation of A (x) will produce the same graph as g(x) is,
⇒ f (x) = f (x + 1)
Learn more about the mathematical expression visit:
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Round each mixed number to the nearest whole number. Then, estimate the product.
870 x 14
_______
Answers
Answer:
12180
Step-by-step explanation:
What is the name of the mountain range that separates Europe from Asia?
Answers
pls help asap 50 pts right answer FOR SURE
When the sun is 45 degrees above the horizon and a tree's shadow is 10ft long, how tall is the tree? *
5ft
7ft
10ft
15ft
Answers
Answer:
C
Step-by-step explanation:
Please refer to the attachment below.
So, we essentially have a 45-45-90 triangle.
In the 45-45-90 triangle, the two legs are equivalent.
Therefore, since one of the legs is 10ft, the other leg, the height of the tree, must also be 10ft.
Hence, our answer is C.
What is the slope ? Will give brainlist
Answers
Answer:
slope = 40
Step-by-step explanation:
slope = rise over run
40 rise for 1 run
5. John and Clark have a lawn moving
companyJohn can mow 3 average sized
lawns in 4 hours. Clark can mow 5 average
sized lawns in 6 hours. Clark argues that since
he is faster, he should get paid more. Do you
agree with Clark? Defend your thinking.
Answers
clark is faster
Step-by-step explanation:
Answer:
Clark is faster
Step-by-step explanation:
If y=10 when x=-4, find y when x=12
Answers
Answer:
-30
Step-by-step explanation:
12/-4 = -3 multiply that by 10 and you'll get -30
A refrigerator costs $799. If the tax is 6.5%, the additional amount of money necessary to purchase the refrigerator is $51.94
True or false
Answers
Answer:
True
Step-by-step explanation:
6.5 x 799 divide 100 = 51.935 = 51.94
Which is the graph of −2x+y+4<9?
HELP PLEASE!
Answers
Answer:
b kno its not right but crown me lol
Step-by-step explanation:
[tex] \frac{ \sqrt[3]{8} }{27} [/tex]
Answers
Answer:
0.0740R
Step-by-step explanation:
The Number is repeating.
To solve the inequality StartFraction m over negative 7 EndFraction less-than-or-equal-to 14, what should be done to both sides?
______________________________________
A) Divide both sides by 14.
B) Divide both sides by –7.
C) Multiply both sides by 14.
D) Multiply both sides by –7.
______________________________________
Answers
Hope it helps u!!!!! ❤️
Answer:
D
Step-by-step explanation:
Jason solved for the quotient of 133.5 ÷ 5.
5 StartLongDivisionSymbol 133.5 EndLongDivisionSymbol minus 10 = 33. 33 minus 30 = 35. 35 minus 35 = a remainder of 0 and quotient of 267. Answer: 133.5 divided by 5 = 267.
What was Jason’s error?
Jason did not multiply correctly.
Jason should have moved the decimal before dividing.
Jason did not put the correct decimal in the quotient.
Jason did not make an error.
Answers
Answer:
Hi there! The answer is C
Step-by-step explanation:
I did it on edge 2020! have a nice day!
Answer:
yo
Step-by-step explanation:
C on edgenuinity
Evaluate x+y when x=−4\5 and y=1\3. Write your answer as a fraction or mixed number in simplest form.
Answers
Answer:
When x=−4\5 and y=1\3 we get x+y = -7/15
Step-by-step explanation:
We are given :
x=−4\5 and y=1\3
We need to evaluate : x+y
Solving:
[tex]x+y\\=-\frac{4}{5}+\frac{1}{3}[/tex]
Taking LCM of denominators 5 and 3, we get 15
[tex]=\frac{-4*3+1*5}{15}\\=\frac{-12+5}{15}\\=\frac{-7}{15}[/tex]
So, when x=−4\5 and y=1\3 we get x+y = -7/15
If a pine tree grows 4 inches per year, how long will it take for the tree to reach a height of 12 feet?
Answers
Answer:
36 years
Step-by-step explanation:
Answer:
36 years hope this helps can i get brainliest
Step-by-step explanation:
You are visiting Baltimore, MD. A taxi company charges a flat fee of $3.00 plus an additional $0.75 per mile.
Write an equation in slope-intercept form that you could use to find the cost of a taxi ride in Baltimore, MD. Let x represent the number of miles and y represent the total cost.
Answers
Answer:
Step-by-step explanation:
y=3+(3x/4) or y=3+(0,75x)
Find the value of x and show work
Answers
Step-by-step explanation:
35+35+x=180 (sum of interior angles of an triangle is 180
therefore
x=180-70
x=110
What is the sum of the measures of the interior of any triangle
Answers
Answer:
180 Degrees
Step-by-step explanation:
Answer:
I think the measure is 180⁰ I know this much only
What is the surface area?
Answers
Answer:
169.4 square units.
Step-by-step explanation:
7*4 = 28
28*3 = 84
B*H/2 is the formula for an equilateral triangle.
7*6.1 = 42.7*2 = 85.4
84+85.4 = 169.4
The surface area of your equilateral triangular prism is 169.4 square units.
Of the 2,374 pieces produced by a particular machine, 27 were defective. What
percent were defective? Round to the nearest hundredth of a percent.
Answers
Answer:
1.14%Step by step explanation
Step one:
The total pieced produces is 2374
total defective is 27
to find the percentage of the defective pieces, we would need to divide the total pieces of the defective by the total number of pieces produced by the machine and multiply the result by 100
Step two
% defecive= (27/2374)*100
% defecive= 0.0113*100
% defecive= 1.137%
To the nearest hundredth of a percent 1.14%
PLS HELP ASAP !!! HONORS ALGEBRA
Solve f(x) = - 5
Answers
Answer:
it will be a straight line side to side, on the -5 y
Answer:
x=-1
Step-by-step explanation:
f(x) = 2x - 3
f(x)=-5
-5=2x-3
-2=2x
x=-1
1.Calculate the Interest Rate
Interest: $38.88
Principal: $72
Time: 3 years
a. 18
b. 19.2%
c. 24%
d. 26.1%
Answers
Answer:
a. 18%
Step-by-step explanation:
Interest: $38.88
Principal: $72
Time: 3 years
Interest rate = x
72*3x = 38.88216x = 38.88x = 38.88/216x = 0.18 = 18%Correct choice is a. 18%
A water faucet leaked 2 ⅙ liters of water every ⅓ of an hour. How many liters did it leak every hour?
Answers
Answer:
6 and 1/2 liters
Step-by-step explanation:
2 times 3 then 1/6 times 3
Given the graph what is the zero of the function
Answers
Answer:
x = 2
Step-by-step explanation:
The zero of the function is the value of x on the x- axis where the graph crosses.
Thus the zero is x = 2
Answer:
X=2
Step-by-step explanation:
I think the zero of the function is when y=0, and on graph we can see that the only point where y=0, x=2.
But don't take my word for it
