Answers
Answer:
dy/dx = - (y/x)^4
Step-by-step explanation:
We are given the equation:
[tex] \displaystyle \large{ \frac{1}{ {x}^{3} } + \frac{1}{ {y}^{3} } = {a}^{3} }[/tex]
Recall important differential formulas such as:
[tex] \displaystyle \large{ \frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)} \\ \displaystyle \large{ \frac{d}{dx} k = 0 \: \: \tt(k \: \: is \: \: a \: \: constant.)}[/tex]
Above is a basic property of differential. Looking at the equation above, differentiate both sides with respect to x. (dy/dx)
[tex] \displaystyle \large{ \frac{d}{dx}( \frac{1}{ {x}^{3} } + \frac{1}{ {y}^{3} } )= \frac{d}{dx} {a}^{3} }[/tex]
Apply the distribution property in.
[tex] \displaystyle \large{ \frac{d}{dx} \frac{1}{ {x}^{3} } + \frac{d}{dx} \frac{1}{ {y}^{3} } = \frac{d}{dx} {a}^{3} }[/tex]
We treat x^3 and y^3 as function to one another, this type of differential will be called implicit once another variable of function is involved f(x,y).
But we treat a^3 as a constant so we just put 0.
[tex] \displaystyle \large{ \frac{d}{dx} \frac{1}{ {x}^{3} } + \frac{d}{dx} \frac{1}{ {y}^{3} } = 0 }[/tex]
Recall the quotient rules for the numerator = 1.
[tex] \displaystyle \large{ \frac{d}{dx} (\frac{1}{f(x)}) = -\frac{ \frac{d}{dx} (f(x))}{ {(f(x))}^{2} } }[/tex]
Thus:
[tex] \displaystyle \large{ - \frac{ \frac{d}{dy} ( {x}^{3} )}{ ({x}^{3})^{2} } + \frac{d}{dx} \frac{1}{ {y}^{3} } = 0 } \\ \displaystyle \large{- \frac{ 3 {x}^{2} }{ {x}^{6} } + \frac{d}{dx} \frac{1}{ {y}^{3} } = 0 }[/tex]
Make sure to recall power rules for differential and exponent laws.
From 3x^2 / x^6.
[tex] \displaystyle \large{ - \frac{ 3 }{ {x}^{4} } + \frac{d}{dx} \frac{1}{ {y}^{3} } = 0 }[/tex]
Now what do we do with 1/y^3? We just differentiate it normally but we multiply with dy/dx once we differentiate the y function.
[tex] \displaystyle \large{- \frac{ 3 }{ {x}^{4} } -\frac{ \frac{d}{dx}( {y}^{3}) }{ ({y}^{3})^{2} } = 0 } \\ \displaystyle \large{ -\frac{ 3 }{ {x}^{4} } - \frac{ 3 {y}^{2} }{ {y}^{6}} = 0 } \\ \displaystyle \large{ - \frac{ 3 }{ {x}^{4} } - \frac{ 3 }{ {y}^{4}} \frac{dy}{dx} = 0 }[/tex]
Solve for dy/dx, add both sides by 3/x^4.
[tex] \displaystyle \large{ - \frac{ 3 }{ {x}^{4} } + \frac{3}{ {x}^{4} } - \frac{ 3 }{ {y}^{4}} \frac{dy}{dx} = 0 + \frac{3}{ {x}^{4} } } \\ \displaystyle \large{ - \frac{ 3 }{ {y}^{4}} \frac{dy}{dx} = \frac{3}{ {x}^{4} } } \\ [/tex]
Divide both sides by - 3/y^4.
[tex] \displaystyle \large{ \frac{- \frac{3}{ {y}^{4} } \frac{dy}{dx} }{ - \frac{3}{ {y}^{4} } } = \frac{ \frac{3}{ {x}^{4} } }{ - \frac{3}{ {y}^{4} } } } \\ \displaystyle \large{ \frac{dy}{dx} = \frac{3}{ {x}^{4} } \times (-\frac{ {y}^{4} }{3} )} \\ \displaystyle \large{ \frac{dy}{dx} = \frac{1}{ {x}^{4} } \times (- \frac{ {y}^{4} }{1}) } \\ \displaystyle \large{ \frac{dy}{dx} = - \frac{ {y}^{4} }{ {x}^{4} } } \\ \displaystyle \large{ \frac{dy}{dx} = - ( { \frac{y}{x} })^{4}}[/tex]
Related Questions
Mary has 3 more apples than Ted and 3 times as many apples as Alex. How many apples do all three of them have combined if Mary has 24 apples
Answers
Answer:
52
Step-by-step explanation:
24-3=21 21÷3=7 24 + 21 + 7 = 52
Find the sum of −6x^2 −1 and x+9
Answers
Answer:
[tex]6 {x}^{2} - 1 + x + 9 \\ = 6 {x}^{2} + x + 8[/tex]
Roxanne is using successive approximation to solve this equation.
She started from a graph, where she found the solution to be between and 0. Using the lower and upper bounds from the graph, Roxanne started the first iteration of successive approximation. Where did Roxanne make a mistake, and what was the error?
Answers
Answer:
B. Roxanne made a mistake at step 5. She should have used x - 1/2 as the new upper bound.
Step-by-step explanation:
Based on the successive approximation method, Roxanne made a mistake at step 5. She should have used x = -1/2 as the new upper bound.
What is successive approximation?In Science, successive approximation can be defined as a classical method that is used in Calculus for solving integral equations or initial value problems.
In this scenario, Roxanne started the first iteration of successive approximation by using the lower and upper bounds of the graph. However, we can deduce that Roxanne made a mistake at step 5 because she should have used x = -1/2 as the new upper bound.
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what is the slope of the line that contains the points (7, -1) and (-2, 2) ?
A - 1/3
B 1/3
C - 3
D - 2
Answers
Step-by-step explanation:
m = [tex]\frac{y_{2}- y_{1} }{x_{2} -x_{1} }[/tex]
m = [tex]\frac{2-(-1)}{(-2)-7}[/tex]
m = [tex]\frac{3}{-9}[/tex]
m = [tex]-\frac{1}{3}[/tex]
Please give me a brainliest answer
Sean has a collection of combs. Each comb is worth $1.29. There are 15 total combs.
What is the total value of his collection?
Answers
Answer:
1.29 x 15 = $19.35
The total value of Sean's collection is $19.35
**hope this helps
Step-by-step explanation:
A line passes through the points (0,5) and (3,6). What is its equation in slope-intercept
form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
Answer:
Slope intercept form:
y = 1/3x + 5
There are 32 students in the 6th grade at Garden Valley Middle School. Eight of them have freckles. What is the ratio of 6th graders with freckles to the total number of 6th graders?
A.
1:4
B.
4:1
C.
1:3
D.
3:1
Answers
Answer:
A. 1:4 is the answer of this question
Answer:
a
Step-by-step explanation:
there's a pattern that goes along with the 32 students in the 6th grade. since 8 is a factor (or multiple?) of 32, and 8x4 is 32, we can simplify.
so, 8 is the first multiple of 32 and 32 is the fourth. if we simplify that, we get 1:4.
the answer is 1:4.
Engineering An overpass intersects two
lanes of a highway. What must the value of x
be to ensure the two lanes are parallel?
Answers
Answer:
Look down :)
Step-by-step explanation:
The two equations must equal 180 degrees. So what you could do it 4x + (2x + 12) = 180
then solve. If this isnt enough information just comment :)
If no I hope this helped
For the two lanes to be parallel, the value of x must be 28 degrees,
The given information states that the same-side interior angles are 4x degrees and (2x + 12) degrees.
For parallel lines, the sum of these angles must equal to 180 degrees.
As per the concept of linear sum,
4x + (2x +12) = 180
Adding the like terms, we get:
(4x + 2x) + 12 = 180
Simplifying, we get:
6x + 12 = 180
Subtract 12 to both sides, and we get:
6x = 180 - 12
Simplifying, we get:
6x = 168
x = 28
Therefore, the value of x is 28 degrees.
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I have to write something here to send a question)
Answers
Answer:
how many children do you have?
Step-by-step explanation:
you said ask a question
Jayson started with 200 anger units. Each week he lost 7 anger units. Write an equation that models Jayson's anger after t weeks.
Answers
Answer:
200 - 7t
Step-by-step explanation:
In order to make an equation, you have to figure out what needs to be put into it. Because Jason starts off with 200 anger units, that number will be subtracted by whatever amount he loses or gains per week, which would be 7. Then, in order to model the number of weeks, you would multiply the variable t to 7, so that it represents how many weeks Jason is going to lose 7 anger units. Therefore, our final equation should be:
Anger units = 200 - 7t
Hope this helps :)
An inlet pipe on a swimming pool can be used to fill the pool in 15 hours. The drain pipe can be used to empty the pool in 20 hours. If the pool is [tex]\frac{1}{3}[/tex] filled and then the inlet pipe and drain pipe are opened, how many more hours would it take to fill the pool? Round your answer to two decimal places, if needed.
_____________ hours
Answers
Answer:
40 hoursStep-by-step explanation:
Rate of filling is 1/15Rate of draining is 1/20If both lines are open the rate of filling is:
1/15 - 1/20 = 4/60 - 3/60 = 1/60 per hourSince tank is 1/3 full we need to fill the remaining 2/3 of it.
Time required:
2/3 : 1/60 = 2/3*60 = 40 hourswhat is proportion mean
Answers
Answer:
a part, share, or number considered in comparative relation to a whole.
Step-by-step explanation:
hope it helps
mark me brainliest pls
Please help!
In the figure below, P is the midpoint of LM, and Q is the midpoint of MN. Use the Midpoint formula to find the location of point P and point Q.
Answers
Answer:
P= (5.5, 8)
Q=(8.5,5)
Step-by-step explanation:
Coordinates of L: (2,7)
Coordinates of M: (9,9)
Coordinates of N: (8,1)
Midpoint formula is: [tex]\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)[/tex]
P is the midpoint of LM, so the coordinates of P will be ((2+9)/2,(7+9)/2)
P's coordinates are (5.5, 8)
Q is the midpoint of MN, so the coordinates of Q will be ((9+8)/2,(9+1)/2)
Q's coordinates are (8.5,5)
Leah hangs 6 posters in a row, each 14 inches wide, on a bedroom wall. There are 4 inches between each poster. For the posters at each end, there are 3 inches between each poster and the end of the wall. What is an appropriate strategy for Leah to use to find the width of the wall?
Answers
Answer:
62 sq in of wall is covered
Step-by-step explanation:
6 x 14 = 84 -22 = 62
this is for the total
4 x 4 = 16
this is for the inches of the inside wall
2 x 3 = 6
this is for the inches of the outside wall
Leah can draw a picture showing each poster and space. Then she can add the measurements together to find the width of the wall option (C) is correct.
What is a rectangle?It is defined as the two-dimensional geometry in which the angle between the adjacent sides is 90 degrees. It is a type of quadrilateral.
It is defined as the area occupied by the rectangle in two-dimensional planner geometry.
The area of a rectangle can be calculated using the following formula:
Rectangle area = length x width
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
Leah must add the widths of all the places and posters to get the breadth of the wall, so the best course of action is:
Leah can draw a picture showing each poster and space. Then she can add the measurements together to find the width of the wall.
Thus, the appropriate strategy for Leah to use to find the width of the wall is Leah can draw a picture showing each poster and space. Then she can add the measurements together to find the width of the wall option (C) is correct.
Learn more about the rectangle here:
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Can you help me plz.
Answers
Answer:
Step-by-step explanation:
pls gimme brainliest if im correct
Answer:
7.77, 7[tex]\frac{4}{9}[/tex], 7.4, 7[tex]\frac{3}{4}[/tex]
Step-by-step explanation:
convert the fractions to decimals
[tex]\frac{4}{9}[/tex] = .44 so 7.44
[tex]\frac{3}{4}[/tex] = .25 so 7.25
7.77, 7.44, 7.40, 7.25
they look to already be in order.
have a lovely day <3
For the simple linear equation below, what is the y-coordinate of the ordered pair when the x-coordinate is -15? y=4/5x Enter your answer in the box.(-15, )
Answers
Answer:
The answer is -12
Step-by-step explanation:
solve algebraically 2x^2+4x-85 ≤-4x+5
Answers
Answer:
x ≤ -9 and x ≤ 5
Step-by-step explanation:
2x² + 4x - 85 ≤ -4x +5
2x² + 8x - 85 ≤ 5
2x² + 8x - 90 ≤ 0
x² + 8x -180 ≤ 0
(x + 18)(x - 10) ≤ 0
(x + 9)(x - 5) ≤ 0
x + 9 ≤ 0
x ≤ -9
x - 5 ≤ 0
x ≤ 5
please help with the mathematical questions
Answers
A = (7x9) + (18x17) = 369
2. P = 16+22+11+7+9+11 = 76
A = (16x11)+(11x15) = 341
3. P = 10+14+29+24+14+19 = 110
A = (24x19) + (14x14) = 652
4. P= 10+17+15+7+9+8 = 66
A=(15x10)+(8x9) = 222
5. P = 13+16+19+10+26+32 = 116
A = (13x10)+(32x16) = 642
6. P = 32+31+19+20+12+12 = 126
A = (31x12)+(20x12)= 612
1. find the measure of each angle
Answers
Step-by-step explanation:
adding both equations together makes the straight line that is QRS. straight lines are 180
you should picture a windshield wiper going side to side and you're trying to see how much farther it has to move until it straightens out at the bottom
(3x-5) + (10x-7) = 180
Write an equation of each parabola in vertex form vertex : (7, 4) point: (5, 16)
Answers
Answer:
y=3(x-7)^2+4
Step-by-step explanation:
two less than seven times a number c
btw this is write the phrase as an algebraic expression
Answers
Answer:
3c-2
Step-by-step explanation:
Hope it helps!!!!!!!
A sales person at a car dealership earns a monthly salary of $2400 and $250 for each vehicle sold
3. The car dealership sets a goal for each salesperson to sell at least 12 cars per month. The salesperson
achieves this goal every month for the entire year. What is the minimum amount of money the
salesperson earns for the entire year? Show your work.
I
Answers
Answer:
8 vehicles
Step-by-step explanation:
Given: Monthly salary of salesperson is $2400
Incentive of salesperson is $250 per vehicle sold.
First, lets compute income to earned apart from salary.
∴ Deduct salary from the target amount
$2000 need to be earned only through incentive, which is $250 per vehicle sold.
Now, solving to get number vehicles to be sold to earn $2000 as incentive.
Number of vehicle=
8 vehicles need to be sold by salesperson to get total earning as $4400.
An entrepreneur wants to rent
a billboard at least 30 feet tall to display an
advertisement for her business. She is 5 feet 6 inches
tall and casts a shadow 7 feet long. At the same
time, a billboard casts a shadow 35 feet long. Is the
billboard tall enough?
Answers
Is 9 a solution to the equation?
3x - 7=20
Yes no
Answers
Answer:
yes
Step-by-step explanation:
because its right i know its right
Jessica has 1 1 week's worth of pay in a wallet and 1 2 week's of pay in the bank.
How many weeks of pay does Jessica have?
Answers
Hello!
11 weeks plus 12 weeks equals the total number of weeks.
23 weeks
Hope this helps!
#HaveAnAmazingDay
[tex]GraceRosalia :)[/tex]
what is 2 over 5 + 3 over 7=
Answers
Answer:
[tex]\frac{29}{35}[/tex] Is correct answer
Step-by-step explanation:
Step 1: We still have different denominators (bottom numbers), though, so we need to get a common denominator. This will make the bottom numbers match. Multiply the denominators together first. Now, multiply each numerator by the other term's denominator.
Now we multiply 2 by 7, and get 14, then we multiply 5 by 7 and get 35.
Now for the second term. You multiply 3 by 5, and get 15, then multiply 5 by 7 and get 35.
Step 2: Since our denominators match, we can add the numerators.
14 + 15 = 29
That gives us an answer of = [tex]\frac{29}{35}[/tex]
In pic
(Hope this helps can I pls have brainlist (crown)☺️
Name the jelly like fluid that the other organelles float around in a cell. O Cytoplasm O Cell Membrane O Chloroplast
Answers
Step-by-step explanation:
[tex] \bf \underline\red{Question}[/tex]
Name the jelly like fluid that the other organelles float around in a cell?
[tex] \bf \underline\red{Answer}[/tex]
Cytoplasm.
Answer:
cytoplasm
step by step explanation
According to the Rational Root Theorem, which statement about f(x)=8x3−4x2+6x+3 is true?
Group of answer choices
Any rational root of f(x) is a multiple of 8 divided by a multiple of 3.
Any rational root of f(x) is a factor of 8 divided by a factor of 3.
Any rational root of f(x) is a multiple of 3 divided by a multiple of 8.
Any rational root of f(x) is a factor of 3 divided by a factor of 8.
Answers
The rational root theorem is used to determine the possible roots of a function.
The true statement is: (d) Any rational root of f(x) is a factor of 3 divided by a factor of 8.
The function is given as:
[tex]\mathbf{f(x) = 8x^3 - 4x^2 + 6x + 3}[/tex]
The possible roots of a function
[tex]\mathbf{f(x) = qx^n + ax^{n-1} +.....+ p}[/tex]
are
[tex]\mathbf{Roots = \pm\frac{Factors\ of\ p}{Factors\ of\ q}}[/tex]
By comparison: p = 3 and q = 8
So, we have:
[tex]\mathbf{Roots = \pm\frac{Factors\ of\ 3}{Factors\ of\ 8}}[/tex]
Hence, the true option is (d) Any rational root of f(x) is a factor of 3 divided by a factor of 8.
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Thembi leaves school at 07:10 to get to school at 07:45. How long does it take her to school and home each week?
Answers
1) In how many ways can a girl choose a two-piece outfit from 9 blouses and 8 skirts?
A) 17
B) 19
C) 144
D) 72
Answers
The total number of ways a girl can choose a two-piece outfit from 9 blouses and 8 skirts is 72.
What is permutation?Permutation is a mathematical calculation of the number of ways a particular set can be arranged, where order of the arrangement matters.
Permutation Formula[tex]^{n} P_{r} = \frac{P!}{(n-r)!}[/tex]
Where,
[tex]^{n}P_{r}[/tex] is permutation
n is the number of objects
r is the number of objects selected.
According to the given question.
Total number of blouses = 9
Total number of skirts = 8
So,
the number of ways of choosing one blouse from 8
=[tex]^{8} P_{1}[/tex]
= [tex]\frac{8!}{(8-1)!} \\[/tex]
= [tex]\frac{8!}{7!}[/tex]
= [tex]\frac{8\times 7!}{7!}[/tex]
= [tex]8[/tex]
And the number of ways of choosing one skirt from 9
[tex]^{9} P_{1} \\= \frac{9!}{(9-1)!} \\= \frac{9\times 8!}{8!} \\=9[/tex]
Therefore,
The total number of ways of choosing two piece outfit from 9 blouses and 8 skirts
= [tex]^{8} P_{1} \times ^{9} P_{1}[/tex]
= [tex]8 \times 9[/tex]
[tex]= 72[/tex]
Hence, the total number of ways a girl can choose a two-piece outfit from 9 blouses and 8 skirts is 72.
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