Answers
Answer:
[tex]\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}[/tex]
General Formulas and Concepts:
Pre-Algebra
Equality PropertiesAlgebra I
FunctionsFunction NotationExponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Algebra II
Logarithms and Natural LogsLogarithmic Property [Multiplying]: [tex]\displaystyle log(ab) = log(a) + log(b)[/tex]Logarithmic Property [Exponential]: [tex]\displaystyle log(a^b) = b \cdot log(a)[/tex]Calculus
Derivatives
Derivative Notation
Derivative 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]
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Logarithmic Derivative: [tex]\displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}[/tex]
Implicit Differentiation
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = x\sqrt[3]{1 + x^2}[/tex]
Step 2: Rewrite
[Equality Property] ln both sides: [tex]\displaystyle lny = ln(x\sqrt[3]{1 + x^2})[/tex]Logarithmic Property [Multiplying]: [tex]\displaystyle lny = ln(x) + ln(\sqrt[3]{1 + x^2})[/tex]Exponential Rule [Root Rewrite]: [tex]\displaystyle lny = ln(x) + ln \bigg[ (1 + x^2)^\bigg{\frac{1}{3}} \bigg][/tex]Logarithmic Property [Exponential]: [tex]\displaystyle lny = ln(x) + \frac{1}{3}ln(1 + x^2)[/tex]Step 3: Differentiate
ln Derivative [Implicit Differentiation]: [tex]\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx} \bigg[ ln(x) + \frac{1}{3}ln(1 + x^2) \bigg][/tex]Rewrite [Derivative Property - Addition]: [tex]\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{d}{dx} \bigg[ \frac{1}{3}ln(1 + x^2) \bigg][/tex]Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{1}{3}\frac{d}{dx}[ln(1 + x^2)][/tex]ln Derivative [Chain Rule]: [tex]\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \frac{d}{dx}[(1 + x^2)][/tex]Rewrite [Derivative Property - Addition]: [tex]\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \bigg( \frac{d}{dx}[1] + \frac{d}{dx}[x^2] \bigg)[/tex]Basic Power Rule]: [tex]\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot (2x^{2 - 1})[/tex]Simplify: [tex]\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot 2x[/tex]Multiply: [tex]\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{2x}{3(1 + x^2)}[/tex][Multiplication Property of Equality] Isolate y': [tex]\displaystyle y' = y \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg][/tex]Substitute in y: [tex]\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg][/tex][Brackets] Add: [tex]\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{5x^2 + 3}{3x(1 + x^2)} \bigg][/tex]Multiply: [tex]\displaystyle y' = \frac{(5x^2 + 3)\sqrt[3]{1 + x^2}}{3(1 + x^2)}[/tex]Simplify [Exponential Rule - Root Rewrite]: [tex]\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Implicit Differentiation
Book: College Calculus 10e
Related Questions
**Help Needed**
The picture below is a sketch of a picnic area with a courtyard. How much of the courtyard is available for use?
Answers
Answer:
87 m²
Step-by-step explanation:
The picnic area is the white inner rectangle, while the courtyard is the outer pink rectangle.
The area of the courtyard available for use = area of the pink rectangle - area of the white rectangle
✔️Area of the courtyard (pink rectangle) = L*W
L = 13 m
W = 7 m
Area of pink rectangle = 13*7 = 91 m²
✔️Area of the picnic (white rectangle) = L*W
L = 4 m
W = 1 m
Area = 4*1 = 4 m²
✅Area available in the courtyard for use = 91 m² - 4 m² = 87 m²
Identify the y-intercept in y=-1/3x + 2
Hint: y = mx + b
1. -1/3
2. 2
3. -2/3
4. -2
Answers
Answer:
2
Step-by-step explanation:
y = mx + b
m = slope
b = y-intercept
y = -1/3x + 2
b = 2
Answer:
since negative 1/3 is the slope and x is the point on the x axis , 2 is the y intercept
5x≥25 help th bewbsanks
Answers
To get this answer, we simply divide by five on both sides to isolate x
Find the distance between the two points rounding to the nearest tenth (if necessary).
(8,-3) and (4,6)
Answers
Answer:
9.8
Step-by-step explanation:
x1 = 8 x2 = 4 y1 = -3 y2 = 6
[tex]\sqrt{(x2 - x1)^{2}+(y2 - y1)^{2}[/tex]
[tex]\sqrt{(4 - 8)^{2}+(6 - -3)^{2}[/tex]
= 9.8488
= 9.8
Finnhas$60inasavingsaccount.Theinterestrateis10%peryearandisnotcompounded.Howmuchinterestwillheearnin1year?
Use the formula i = prt, where i is the interest earned, p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.
Answers
Answer:
I = $ 6.00
Step-by-step explanation:
Equation:
I = Prt
Calculation:
First, converting R percent to r a decimal
r = R/100 = 10%/100 = 0.1 per year,
then, solving our equation
I = 60 × 0.1 × 1 = 6
I = $ 6.00
The simple interest accumulated
on a principal of $ 60.00
at a rate of 10% per year
for 1 years is $ 6.00.
Which of these forms a perfect square: Select all expressions that are prefect squares. Group of answer choices (a+3)(3+a) (a+b)(b+a) (2x-1)(2x+1) (x+5)(x+5) (-9+c)(c-9) (y-10)(10-y) (4-3x)(3-4x)
Answers
Answer:
Step-by-step explanation:
x = 2
step-by-step explanation:
if sr parallel to ba then the sides bc and ac are in proportion, that is
\frac{sb}{cs} = \frac{ra}{cr} ← substitute values
\frac{4x+1}{2x} = \frac{13.5}{6} ( cross- multiply )
27x = 6(4x + 1)
27x = 24x + 6 ( subtract 24x from both sides )
3x = 6 ( divide both sides by 3 )
x = 2
The question I’m trying to figure out is What is y?
Answers
Answer:
y = 136
Step-by-step explanation:
x + 54 + 82 = 180
x + 136 = 180
x = 44
x + y = 180
44 + y = 180
y = 180 - 44
y = 136
22% of _ peaches is 11 peaches
Answers
|y-x|,if y>x pls dont write in absolute value
Answers
Answer:
The answer is y-x
Step-by-step explanation:
Applying the rule that |y| = y when y>0, we can plug in the values of y-x instead of y.
So, |y-x| = y-x if y>0
The perimeter of a rectangular plot of land for a new housing development is 270 m.
The width of the site is 60 m.
Calculate the length of the building site.
Answers
The length of the building site is 75 meters.
To calculate the length of the rectangular plot of land, we can use the formula for the perimeter of a rectangle, which is given by:
Perimeter = 2 * (Length + Width)
In this case, the width is given as 60 m, and the perimeter is given as 270 m. We can substitute these values into the equation and solve for the length.
270 = 2 * (Length + 60)
Dividing both sides of the equation by 2 gives:
135 = Length + 60
Subtracting 60 from both sides of the equation gives:
Length = 135 - 60
Length = 75
It's important to note that the length of the building site is specific to this particular rectangular plot of land with a given width and perimeter. If the width or perimeter were to change, the length would also change accordingly. Additionally, this calculation assumes that the shape of the plot is a rectangle and that the perimeter only includes the sides of the plot, not any internal structures or obstacles.
For more such questions on length
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One third is equivalent to what number over 12?
Answers
Answer:
4/12
Step-by-step explanation:
1/3=x/12
using the butterfly method you see that
12=3x
and when solved you get
x=4
giving you
4/12
Also multiply 1/3 by 4 on each side
One third is equivalent to 4 over 12.
What is the fraction?In Mathematics, fractions are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
Let the unknown number be x.
Here, 1/3 = x/12
Multiply both the numerator and denominator of 1/3 by 4, we get
4/12
So, 4/12 = x/12
x=12
Therefore, the unknown number is 12.
To learn more about the fraction visit:
brainly.com/question/1301963.
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a triathlon event in which participants run,bike, and swim certain distances is mapped out in a triangle as shown below.
Answers
Answer:
A
Step-by-step explanation:
The statement 'the bike distance is greater than the run distance.' must be true.
The correct answer is option D.
What is distance?"It is the length of the line joining the two points."
Angle properties:In any triangle,
the smallest side and smallest angle are opposite one another. the mid-sized side and mid-sized angle are opposite one another.the largest side and largest angle are opposite one another.For given example,
We have given two angles of triangle.
65° and 57°
We know, the sum of all angles of triangle is 180°.
Let 'x' be the remaining angle.
⇒ x + 65° + 57 = 180°
⇒ x +122° = 180°
⇒ x = 58°
We know,
In any triangle,
- the smallest side and smallest angle are opposite one another.
- the mid-sized side and mid-sized angle are opposite one another.
- the largest side and largest angle are opposite one another.
Here,
57° < 58° < 65°
⇒ Run < Bike < Swim
This means, the statement 'the bike distance is greater than the run distance.' is correct.
The correct answer is option D.
Learn more about angle here:
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https://studio.code.org/projects/flappy/cYGc2S_X9-cgWtBUmJ5BJta5xku7iqyon0wzrKhC-Do
Answers
Answer:
so fun and amazing!
Step-by-step explanation:
ok soo ima need help !! {re-posting this one}
question-
Select all of the ratios that are equivalent to 6:16.
2:8
3:8
12/18
30/80
18 to 48
Answers
Answer:
3:8
30/80
18 to 48
Step-by-step explanation:
3:8 (6 and 16 divided by 2)
30/80 (6 and 16 times 5)
18 to 48 (6 and 16 times 3)
What is X in this equation?
Answers
Answer:
B) 44
Step-by-step explanation:
x + 54 + 82 = 180
x + 136 = 180
x = 180 - 136
x = 44
what are 4 major accomplishments in Will Smith's life?
Answers
Answer:
He first rapped as a duo named Dj Jazzy Jeff and the fresh prince in 1988, with the hit songs Summertime and Parents just don't understand
He then started his career as an actor with the hit sitcom The fresh prince of Bel-air in 1990
He then later gained more popularity in 1997 with hit songs like Gettin jiggy and Miami
He became a well-known actor for movies like Bad boys, Men in black, and the Aladin remake in 2019
Step-by-step explanation:
Hope this helps and don't forget to follow!
Pls pick me as Brainliest!
Nicole received the following scores on her last 10 math assignments.
Answers
Answer: 89
Step-by-step explanation:
right on edge
For which of the following are x=4 and x=-4 solutions?
Answers
Answer:
2x2 - 5x - 12 = 0.
(2x + 3)(x - 4) = 0.
2x + 3 = 0 or x - 4 = 0.
x = -3/2, or x = 4. I hope this helps
Step-by-step explanation:
Determine which polynomial is a perfect square trinomial. Explain your answer.
Group of answer choices
A. 49x2 − 28x + 16
B. 9a2 − 30a + 25
C. 25b2 − 45b − 81
D. 16x2 − 24x − 9
Answers
Answer:
The answer is B. 9a2-30a+25
Step-by-step explanation:
First of all, the second sign is going to be +, so you can toss out C and D without checking.
Then you take the square root of the two ends, multiply them, multiply by two and see if it matches.
sqrt(49) * sqrt(16) = 7 * 4 = 28
28 * 2 = 56
56 <> 28
So toss A.
sqrt(9) * sqrt(25) = 3 * 5 = 15
15 * 2 = 30
So the answer is B
(3a - 5)^2 = 9a^2 - 30a + 25
By the way, use ^ to indicate an exponent.
Evaluate the expression for x = 2 and y = 4 . (x ^ 5 * y ^ 7) / (x ^ 3 * y ^ 5) numbers only
Answers
Answer:
Answer is 64.
Step-by-step explanation:
524288/8192 = 64
If 2^2x = 2^3, what is the value of x?
Answers
Answer:
x = 2
Step-by-step explanation:
3/4(5a - 16)- 1/3(6a-3)
Answers
Answer:
If you want an expression answer (Simplify):
= 7/4a - 11
If you want an answer to the variable a (Find 'a'):
7/4a - 11 = 0
7/4a = 11
a = 44/7
Step-by-step explanation:
3/4(5a - 16) - 1/3(6a - 3)
= 15/4a - 12 - 2a + 1
= 7/4a - 11
Cassie paid $276.45 for 5 months of cell phone service. At that rate, how much will Cassie pay for a full year of cell phone service?
NO ANSWERS FROM UNKNOWN FILES PLEASE OR I'LL REPORT IT!!!
Answers
Find the opposite of: - 4x2 - 6x + 3
Answers
A student purchased 9 binders for $11.07.
Solve the equation to determine the cost, in dollars, for each binder.
$
Answers
Answer:
$1.23
Step-by-step explanation:
9 binders for 11.07, where x = num of binders
9x = 11.07 [Divide both sides by 9]
x = $1.23
Answer:
each binder costs $1.23
Step-by-step explanation:
$11.07 divided by 9 equals $1.23
pls help me i suck at math and i really need this done.
Answers
Answer:
1.x²+x-2
=x²+2x-x-2
=x(x+2)-1(x+2)
=(x+2)(x-1)
2.
(x²+x-2)/(x-1)
={x²+2x-x-2}/(x-1)
={x(x+2)-1(x+2)}/(x-1)
={(x+2)(x-1)}/(x-1)
=(x+2) true
Using the same scenario as above, but now you do not know the population variance. You collect a random sample of n = 21 cereal boxes and find the sample mean is 12.8 and the sample standard deviation to be 0.5. Test the null hypothesis H0: µ = 13 against the alternative hypothesis HA: µ < 13 at level of significance α = 0.01.
Required:
a. Find the test statistic.
b. Using the rejection region, do you reject or not reject the null hypothesis?
c. Find the p-value.
d. Using the p-value, do you reject or not reject the null hypothesis?
e. In context of your problem, what is your conclusion about the ounces contained in cereal boxes?
Answers
Answer:
-1.833 ; we do not reject the Null.
Pvalue = 0.0409 ; We do not reject the Null
There is no significant evidence to conclude that the number of ounces contained in the cereal boxes is less than 13
Step-by-step explanation:
H0 : μ = 13
HA : μ < 13
Test statistic :
(xbar - μ) ÷ s/sqrt(n)
(12.8 - 13) ÷ 0.5/sqrt(21)
-0.2 / 0.1091089
= - 1.833
The critical value at α = 0.01 ; df = 21 - 1 = 20 = 2.528
-1.833 > - 2. 528 (left tailed test) ; Hence we do not reject the Null.
Test statistic :
Pvalue from Test statistic at DF = 20 ; α = 0.01
Pvalue = 0.0409
Pvalue > α
0.0409 > 0.01 ; Hence, we do not reject H0.
There is no significant evidence to conclude that the number of ounces contained in the cereal boxes is less than 13
questions number 1
How much paper is used for the label of the container, which covers the lateral surface area of the cylinder? Round your answer to the nearest tenth.
question number 2
what volume of oats will fill the container? Round your answer to the nearest tenth.
Answers
9514 1404 393
Answer:
145.3 square inches
Step-by-step explanation:
The lateral area of the cylinder is given by ...
LA = 2πrh
LA = 2π(2.5 in)(9.25 in) = 46.25π in²
LA ≈ 145.3 in²
About 145.3 square inches of paper is used for the label.
In AFGH, FG – HF and mZF = 56°. Find m_H.
Answers
Answer:
m<H = 62°
Step-by-step explanation:
If FG is congruent to HF, it means their opposite angles are also congruent to each other. This implies that, <H = <G. Thus, ∆FGH is an isosceles triangle with base angles <F and <G.
Given that m<F = 56°, therefore:
56° + 2(m<H) = 180° (sum of triangle)
2(m<H) = 180° - 56°
2(m<H) = 124°
m<H = 124/2
m<H = 62°
X^3 - 5x^2 + 2x + 8 = 0; x - 2 factor and find all roots
Answers
Answer:
The zeroes are -1, 2, and 4. Factored: (x+1)(x-4)(x-2)=0
Step-by-step explanation: