Answer:
[tex]\displaystyle \frac{dy}{dx} = 3(x + 8)^{3x} \bigg[ \ln (x+ 8) + \frac{x}{x + 8} \bigg][/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle f(x) = (x + 8)^{3x}[/tex]
Step 2: Differentiate
Exponential Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \frac{d}{dx} \bigg[ 3x \ln (x + 8) \bigg][/tex]Derivative Rule [Product Rule]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ \frac{d}{dx}(3x) \ln (x + 8) + 3x \frac{d}{dx}[ \ln (x + 8)] \bigg][/tex]Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ 3 \frac{d}{dx}(x) \ln (x + 8) + 3x \frac{d}{dx}[ \ln (x + 8)] \bigg][/tex]Derivative Rule [Basic Power Rule]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ 3 \ln (x + 8) + 3x \frac{d}{dx}[ \ln (x + 8)] \bigg][/tex]Logarithmic Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ 3 \ln (x + 8) + \frac{3x}{x + 8} \frac{d}{dx} (x + 8) \bigg][/tex]Rewrite [Derivative Property - Addition/Subtraction]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ 3 \ln (x + 8) + \frac{3x}{x + 8} \Big[ \frac{d}{dx}(x) + \frac{d}{dx}(8) \Big] \bigg][/tex]Derivative Rule [Basic Power Rule]: [tex]\displaystyle \frac{dy}{dx} = (x + 8)^{3x} \bigg[ 3 \ln (x + 8) + \frac{3x}{x + 8} \bigg][/tex]Simplify: [tex]\displaystyle \frac{dy}{dx} = 3(x + 8)^{3x} \bigg[ \ln (x+ 8) + \frac{x}{x + 8} \bigg][/tex]∴ we have found the derivative of the given function.
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Marisa painted one-half of her bedroom in three-fourths of an hour. At this rate, how long would it take her to paint the entire room? three-eighths of an hour eight-thirds of an hour four-sixths of an hour six-fourths of an hour
Answer:
6/4 of an hour hope this helps.
Ally brought 1/2 pound of cashews 5/6 pound of granola.plot the fractions on the number line.which quantity is greater?explain.
Write an explicit formula fo the sequence 7,2,-3,-8,-13.... Then find the 14th term
(please explain)
Answer:
The 14th term is -58
Step-by-step explanation:
The formula for an arithmetic sequence is
an = a1+d(n-1)
where a1 is the first term
d is the common difference and
n is the term number
a1 = 7
d = 2-7 = -5
Check -3-2 = -5 so d = -5
an = 7 -5(n-1)
We want the 14th term
a14 = 7 -5(14-1)
a14 = 7 -5(13)
a14 = 7 -65
a14 = -58
PLLLZ HELP I'LL GIVE BRAINLISET
Answer:
Step-by-step explanation:
Sandy only because the 2 prisms overlap
Solve for the midpoint of the segment with endpoints (5,18) and (-11,2)
Answer:
Midpoint ( -3, 10)
Step-by-step explanation:
To find the midpoint , we use the formula
midpoint = (x1+x2)/2, (y1+y2)/2
Substituting what we know
= (5+-11)/2, (18+2)/2
= (-6/2), (20/2)
=-3,10
NEED HELPPPP ASAP PLEASEEEE
Answer: 35.6 (choice A)
To get this answer, add up the values: 39+42+27+45+25 = 178
Next, divide that result by 5 because there are 5 scores added up in the previous step. Doing this division gets us to: 178/5 = 35.6
what is the following quotient -2 srt8 4+sqrt 12
[tex]\dfrac{-2\sqrt8}{4+\sqrt{12}}=\dfrac{-2\sqrt{4\cdot2}}{4+\sqrt{4\cdot3}}=\dfrac{-2\cdot\sqrt4\cdot\sqrt2}{4+\sqrt4\cdot\sqrt3}=\dfrac{-2\cdot2\sqrt3}{4+2\sqrt3}=\dfrac{-4\sqrt2}{2(2+\sqrt3)}=-\dfrac{2\sqrt2}{2+\sqrt3}\\\\\text{Use}\ (a+b)(a-b)=a^2-b^2\\\\=\dfrac{-2\sqrt2}{2+\sqrt3}\cdot\dfrac{2-\sqrt3}{2-\sqrt3}=\dfrac{-2\sqrt2(2-\sqrt3)}{2^2-(\sqrt3)^2}\\\\\text{Use distributive property and}\ \sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\\\\=\dfrac{-4\sqrt2+2\sqrt6}{4-3}=\dfrac{2\sqrt6-4\sqrt2}{1}=2\sqrt6-4\sqrt2[/tex]
Divide. Give special attention to the order of division. a. 72 ÷ 9 ÷ 2 b. (18 ÷ 6) ÷ 3 c. 45 ÷ 5 ÷ 3 d. 144 ÷ (12 ÷ 2)
Answer:
a. 4
b. 1
c. 3
d. 24
Step-by-step explanation:
a. 72 ÷ 9 ÷ 2
= (72÷9)÷2
= 8÷2
= 4
b. (18 ÷ 6) ÷ 3
= 3÷3
= 1
c. 45 ÷ 5 ÷ 3
= (45÷5)÷3
= 9÷3
=3
d. 144 ÷ (12 ÷ 2)
= 144÷6
= 24
Evaluation the expression when x=-1 and y=3.
5x^-6y^-2
—————
Y
Put the vales of x and y to the expression:
[tex]x=-1;\ y=3\\\\\dfrac{5x^{-6}y^{-2}}{y}=\dfrac{5(-1)^{-6}(3)^{-2}}{3}=\dfrac{5(-1)^6\left(\dfrac{1}{3}\right)^2}{3}=\dfrac{5(1)\left(\dfrac{1}{9}\right)}{3}=\dfrac{\dfrac{5}{9}}{3}\\\\=\dfrac{5}{9}\cdot\dfrac{1}{3}=\boxed{\dfrac{5}{27}}[/tex]
I need help please show work.
Answer:
The midpoint of the segment is (6,5)
Step-by-step explanation:
To find the midpoint of a segment we use the formula
midpoint = (x1+x2)/2, (y1+y2)/2
= (10+2)/2 , (7+3)/2
= 12/2, 10/2
=6,5
The midpoint of the segment is (6,5)
work out the value of (10^4)^2
plsssssss this is urgentttttt
[tex](10^4)^2[/tex]
When you multiply an exponent directly into another exponent, you multiply the exponents.
For example:
[tex](x^{2})^3=x^{2(3)}=x^6[/tex]
[tex](x^5)^3=x^{5(3)}=x^{15}[/tex]
So:
[tex](10^4)^2=10^{4(2)}=10^8=100,000,000[/tex]
Answer:
it is 200000
Step-by-step explanation: ok you want me to show you! huh! :)
so you do 10 times the 4 which is 100000
next you times it by 2 which is equal to 200000
so your answer is 200000
A man bought 8 boxes highlighters, each with 24 highlighter pens inside, for $9 each box. He sold the highlighters for $1 each. Did he earn money or lose money? How much money did he earn or lose?
Answer:
The man earned money:
Step-by-step explanation:
$1 x 24 pieces =$24 revenue per box
$24 - $9 = $15 earning per box
$15 x 8 boxes = $120 total earnings
Use a graphing calculator or online application to find the solution to 3 -x = x 3 + x to the nearest tenth.
The solution is approximately x = a0.
Answer:
x=1
Step-by-step explanation:
Question says to use a graphing calculator or online application to find the solution to [tex]3-x = x^3 + x[/tex] to the nearest tenth.
So we will create two equations y = 3-x and [tex]y = x^3 + x[/tex] using each sides of the given equation.
Now we graph both equation using some online calculator or graphing calculator. I'm going to use Desmos calculator.
Intersection point of both graphs will give the solution.
From graph we see that intersection point is at (1,2).
We are looking for value of x only.
Hence final answer is x=1
Answer:5
Step-by-step explanation:
Create an expression without parentheses that is equivalent to 4(6x + x).
is 0.6 a rational number why or why not
Answer:
.6 is a rational number
Step-by-step explanation:
.6 can be written as 6/10
This is the ratio of integers , which is the definition of rational numbers
Answer:
Rational
Step-by-step explanation:
A rational number is a number that can be expressed as a fraction of integers. .6 (or any terminating decimal) can be expressed as a fraction. .6 is the same as 6/10. Since it is able to be expressed as a fraction, it is a rational number.
Naomi and Sofia go to the movie theater and purchase refreshments for their friends. Naomi spends a total of $111.25 on 9 drinks and 8 bags of popcorn. Sofia spends a total of $33.25 on 3 drinks and 2 bags of popcorn. Write a system of equations that can be used to find the price of one drink and the price of one bag of popcorn. Using these equations, determine and state the price of a bag of popcorn, to the nearest cent.
Answer: popcorn = $5.75 per bag
Step-by-step explanation:
Set up the equation for each person.
Let x represent the cost of one drink and y represent the cost of one bag of popcorn.
drinks popcorn Cost EQUATION
Naomi: 9 8 $111.25 9x + 8y = 111.25
Sofia: 3 2 $33.25 3x + 2y = 33.25
Next, solve the system:
9x + 8y = 111.25 → 1(9x + 8y = 111.25) → 9x + 8y = 111.25
3x + 2y = 33.25 → -3(3x + 2y = 33.25) → -9x - 6y = -99.75
2y = 11.50
y = 5.75
Answer:
System of equations :
$111.25=$9x+$8y
$33.25=$3x+$2y
The cost of 1 drink = $x = $7.25
The cost of 1 popcorn bag = $y = $5.75
Step-by-step explanation:
Given :
Naomi spends a total of $111.25 on 9 drinks and 8 bags of popcorn.
Sofia spends a total of $33.25 on 3 drinks and 2 bags of popcorn.
To Find : A system of equations that can be used to find the price of one drink and the price of one bag of popcorn.
Solution :
Let cost of 1 drink = $x
⇒ cost of 9 drinks = $9x
⇒ cost of 3 drinks = $3x
Let cost of 1 popcorn bag = $y
⇒Cost of 8 popcorn bags = $8y
⇒Cost of 2 popcorn bags = $2y
Since Naomi spends a total of $111.25 on 9 drinks and 8 bags of popcorn.
So, equation will be : $111.25=$9x+$8y --(A)
Since, Sofia spends a total of $33.25 on 3 drinks and 2 bags of popcorn.
So, equation will be : $33.25=$3x+$2y--(B)
SOLVING A and B by using substitution method
finding value of x from equation B :
[tex]x=\frac{33.25-2y}{3}[/tex]
Putting this value in Equation (A)
⇒[tex]111.25=9(\frac{33.25-2y}{3})+8y[/tex]
⇒[tex]111.25=99.75-6y+8y[/tex]
⇒[tex]111.25=99.75+2y[/tex]
⇒[tex]y = 5.75[/tex]
Thus, cost of 1 popcorn bag =y = $5.75
Now put this value of y in equation A to get value of x
⇒$111.25=$9x+$8(5.75)
⇒$111.25-$46 = 9x
⇒$65.25 = 9x
⇒65.25/9 = x
⇒$7.25 = x
Thus, The cost of 1 drink = $x = $7.25
The cost of an online movie rental service is $89.94 for 6 month. What is the cost per month? Please show work
Karin wants to use the distributive property to mentally find the value of 19 x 42+ 19x 58. Which expression can she use?
Answer:
19(42 + 58)
Step-by-step explanation:
Assuming x = multiply, you are trying to solve the distributive property. Divide common factors from both terms (in this case 19).
~
Answer:
19(42 + 58)
Step-by-step explanation:
19 * 42 + 19 * 58 =
= 19 * (42 + 58) <------- this is the expression
= 19 * 100
= 1900
Number 15 please ASAP
With an explanation if possible
Answer:
D 1/40
Step-by-step explanation:
Of the dogs at the show, 1/12 were hounds
Of the hounds, 3/10 were English foxhounds
To determine how many at the show were English foxhounds, we multiply 1/12 by 3/10
1/12* 3/10 = 3/120 = 1/40
the length of a rectangle is 2 cm more than the width. if the area of the rectangle is 24 cm2 what is the length of the rectangle?
The question is about finding the length of a rectangle. By creating and solving a quadratic equation with provided information, the width and length of the rectangle are 4 cm and 6 cm respectively.
Explanation:The problem involves finding the length of a rectangle. To solve this, we need to consider the fact that we know information about the dimensions and area of the rectangle. The area of a rectangle is obtained by performing the multiplication of the length and the width. In this specific problem, it's stated that the length is 2 cm more than the width, and we also know that the area of the rectangle is 24 cm2.
Let's represent the width by x. Therefore, the length would be x+2. Since the area is the length multiplied by the width, we create the equation x*(x+2) = 24, which simplifies to x2+2x=24. Subtracting 24 from both sides, we get x2+2x-24 = 0.
This a quadratic equation and we'll solve for x using the quadratic formula, which results in x = 4 or x = -6. Since the width cannot be negative, x = 4 cm. After this, substitute the width (x) into the length equation. The length (x + 2) is 4 cm + 2 cm = 6 cm.
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The length of the rectangle is [tex]\( 2\sqrt{7} \)[/tex]cm.
To solve this problem, let's denote the width of the rectangle as [tex]\( w \)[/tex] cm. According to the information given, the length of the rectangle is then [tex]\( w + 2 \)[/tex] cm. The area of a rectangle is given by the product of its length and width, so we can set up the following equation:
[tex]\[ w \times (w + 2) = 24 \][/tex]
Expanding the equation, we get:
[tex]\[ w^2 + 2w = 24 \][/tex]
To solve for [tex]\( w \)[/tex], we need to rearrange the equation into standard quadratic form:
[tex]\[ w^2 + 2w - 24 = 0 \][/tex]
Now, we can factor this quadratic equation:
[tex]\[ (w + 6)(w - 4) = 0 \][/tex]
Setting each factor equal to zero gives us two possible solutions for [tex]\( w \)[/tex]:
[tex]\[ w + 6 = 0 \quad \text{or} \quad w - 4 = 0 \] \[ w = -6 \quad \text{or} \quad w = 4 \][/tex]
Since the width of a rectangle cannot be negative, we discard[tex]\( w = -6 \)[/tex] and accept [tex]\( w = 4 \)[/tex] cm as the width of the rectangle.
Now that we have the width, we can find the length by adding 2 cm to the width:
[tex]\[ \text{Length} = w + 2 = 4 + 2 = 6 \text{ cm} \][/tex]
However, we need to check if this length satisfies the area condition:
[tex]\[ \text{Area} = w \times \text{Length} = 4 \text{ cm} \times 6 \text{ cm} = 24 \text{ cm}^2 \][/tex]
This confirms that the length is indeed 6 cm when the width is 4 cm. Therefore, the length of the rectangle is [tex]\( 2\sqrt{7} \)[/tex] cm, since [tex]\( 6 = 2\sqrt{7} \)[/tex] when rounded to the nearest whole number.
Determine whether the lines are parallel. Use slope to explain your answer. Use the graph shown below.
The slope is of the red line is ___
The slope is of the blue line is ___
The lines do/do not have the same slope so they are/are not parallel.
Answer:
red 2/3
blue 1/2
The lines do not have the same slope so they are not parallel
Step-by-step explanation:
We can find the slope of the red line by using the formula
m = (y2-y1)/(x2-x1)
We have the points (0,2) and (3,4)
m= (4-2)/ (3-0)
= 2/3
We can find the slope of the blue line by using the formula
m = (y2-y1)/(x2-x1)
We have the points (0,-2) and (4,0)
m= (0--2)/ (4-0)
= (0+2)/(4-0)
= 2/4
=1/2
The lines do not have the same slope so they are not parallel
Pizza is cut into six equal pieces each slice is then cut in half to create two smaller equal pieces. what fraction of the total pizza does one of the small pieces represent??
Answer:
1/12
Step-by-step explanation:
6 x 2 = 12
6 = the original amount of slices
2 = it was cut in half so multiply by 2
each slice now represents 1/12
Answer:
Each slice represents 1/12th of the total pizza.
Step-by-step explanation:
1 = 1/6 = 1/12
Find the distance between the two points shown in the coordinate plane below to the nearest hundredth.
(-2, -2)
(3, 4)
Answer:
7.81
Step-by-step explanation:
Given points (-2,-2) and (3, 4)
To find distance between points we apply distance formula
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
(x1,y1) is (-2,-2) and (x2,y2) is (3,4)
Plug in the values in the formula
[tex]D = \sqrt{(3-(-2))^2+(4-(-2))^2}[/tex]
[tex]D = \sqrt{(5)^2+(6)^2}[/tex]
[tex]D = \sqrt{25+36}[/tex]
[tex]D = \sqrt{61}[/tex]
D= 7.810249676
Round to nearest hundredth
So distance = 7.81
Nate and Laura picked apples. Laura picked 1/2 as many as Nate picked. Together they picked 90 apples. How many did each of them pick?
Let n stand for the number of apples Nate picked.
Equation:____
How many apples did Nate pick?______
How many apples did Laura pick?______
Answer: Nate picked 60 apples and Laura picked 30. (1/2)x+x=90
Step-by-step explanation:
Population density is the number of people per unit of area. What is the population density of a state that has 1,627,260 people in 1,490 square miles? Round to the nearest whole number
Answer:
= 1092 people/ sq mile
Step-by-step explanation:
Population density is the number of people per unit of area.
We know the population is 1627260
The are is 1490 mi ^2
Population density = 1627260/1490
=1092.120805 people / sq mile
Round to the nearest whole number
= 1092 people/ sq mile
Jamal is shopping with a gift card he received for his birthday. After he purchases two T- shirts the gift card balance has dropped from $50 top $20.02. What is the unit rate of the cost per T- shirt?
Answer:
14.99
Step-by-step explanation:
20.02=50-2t
20.02-50= -2t
-29.98= -2t
Roger is having a picnic for 72 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality gives the minimum number of packages he will need to buy? A) p ≥ 9 B) p ≥ 6
C) p ≤ 6 D) p ≤ 9
Answer:
The answer is A.
Step-by-step explanation:
To solve this problem, use the inverse operation of multiplication, which is division. 72 ÷ 8 = 9. Since the problem stated "which inequality gives the minimum number of packages he will need to buy?" 9 is the minimum number. Therefore, A is the answer.
Show Work:
8p ≥ 72
÷8 ÷8
p ≥ 9
Answer:
A p≥9
Step-by-step explanation:
4. A horse is tethered to one corner of a square plot of side 42 m, by a rope 35 m long. Find i) what area it can graze ii) what area will be left ungrazed.
Answers:
i) It can graze an area of approximately 961.625 square meters
ii) The ungrazed area is roughly 802.375 square meters
note: I used pi = 3.14 which is approximate; if you want more accuracy, use more decimal digits in pi
===============================================
Explanation:
Check out the attached image below for a visual diagram.
The area of a circle is the formula
A = pi*r^2
Use r = 35 as the radius and pi = 3.14 as the approximation for pi, to get,
A = pi*r^2
A = 3.14*35^2
A = 3846.5
Then divide this by 4 to get the area of a quarter circle
A/4 = 3846.5/4 = 961.625
This is the approximate area the horse can graze
------------
Subtract this from 42^2 = 42*42 = 1764 because this is the area of the entire square. Doing this will get you the ungrazed area
Ungrazed area = (entire plot) - (grazed area)
Ungrazed area = (entire area of square) - (area of quarter circle)
Ungrazed area = 1764 - 961.625
Ungrazed area = 802.375
If triangle TOY is equal to triangle STU, then what corresponding side is congruent to side US?
when does the price of an item increase
Final answer:
The price of an item can increase due to inflation, a reduction in supply, or an increased demand. High inflation rates make it challenging to determine the exact causes of price increases and complicate decision-making for both consumers and businesses.
Explanation:
The price of an item can increase due to various factors such as inflation, a decrease in the supply of the good, or an increase in the demand for the good. In situations of high inflation, such as the case in Israel in 1985 with an annual inflation rate of 500%, prices can change so frequently that stores might not post prices directly on items. This makes it challenging for consumers and businesses to respond to economic signals, complicates price comparisons, and makes it difficult to deduce the cause of price increases.
Inflation implies a general increase in prices and can affect all goods and services, leading consumers to spend money quickly before prices rise further. A decrease in supply, perhaps due to production issues or scarcity of resources, can lead to higher prices if demand remains constant. Conversely, an increase in demand for a good without a corresponding increase in supply can also drive prices up. Recognizing the reasons behind rising prices can be complex, especially when high inflation rates obscure the underlying economic indicators.