If z=(x+y)e^y and x=5t and y=1−t^2, find the following derivative using the chain rule. Enter your answer as a function of t.
dz/dt= ?
Answers
By the chain rule, if [tex]z=f(x,y)=f(x(t),y(t))[/tex], then
[tex]\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial f}{\partial x}\dfrac{\mathrm dx}{\mathrm dt} + \dfrac{\partial f}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}[/tex]
We have [tex]f(x,y)=(x+y)e^y[/tex], for which
• [tex]\dfrac{\partial f}{\partial x} = e^y[/tex]
• [tex]\dfrac{\partial f}{\partial y} = e^y + (x+y)e^y = (x+y+1)e^y[/tex]
and [tex]x(t)=5t[/tex] and [tex]y(t)=1-t^2[/tex], so that
• [tex]\dfrac{\mathrm dx}{\mathrm dt} = 5[/tex]
• [tex]\dfrac{\mathrm dy}{\mathrm dt} = -2t[/tex]
Putting everything together, we get
[tex]\dfrac{\mathrm dz}{\mathrm dt} = 5e^y - 2t(x+y+1)e^y \\\\ \dfrac{\mathrm dz}{\mathrm dt} = 5e^{1-t^2} - 2t(5t-t^2+2)e^{1-t^2} = \boxed{\left(2t^3-10t^2-4t+5\right)e^{1-t^2}}[/tex]
The solution of the derivative using the chain rule is [tex]\dfrac{dz}{dt}=(2t^3-10t^2-4t+5)e^{1-t^2}[/tex].
What is the Chain rule?The chain rule explains how to calculate a composite function's derivative.
What is a composite function?If you may write [tex]f(g(x))[/tex] , a function is composite. It's a function within a function, or a function of a function, in other words.
By the chain rule, we can write if [tex]z=f(x,y)=f(x(t),y(t))[/tex], then,
[tex]\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x} \dfrac{dx}{dt}+\dfrac{\partial f}{\partial y} \dfrac{dy}{dt}[/tex]
As it is given to us, [tex]f(x,y)=(x+y)e^y[/tex], therefore, we can write,
[tex]\dfrac{\partial f}{\partial x}=e^y[/tex][tex]\dfrac{\partial f}{\partial y}=e^y+(x+y)e^y=(x+y+1)e^y[/tex]And x(t)=5t, while y(t)=1-t², So,
[tex]\dfrac{dx}{dt}=5[/tex][tex]\dfrac{dy}{dt}=-2t[/tex]Now, Substitute all the values together, in the equation,
[tex]\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x} \dfrac{dx}{dt}+\dfrac{\partial f}{\partial y} \dfrac{dy}{dt}[/tex]
Substituting the values we will get,
[tex]\dfrac{dz}{dt} &=(5 \times e^y)+[-2t(x+y+1)e^y]\\\\[/tex]
[tex]=5e^{1-t^2}-2t(5t-t^2+2)e^{1-t^2}\\\\=(2t^3-10t^2-4t+5)e^{1-t^2}[/tex]
Hence, the solution of the derivative using the chain rule is [tex]\dfrac{dz}{dt}=(2t^3-10t^2-4t+5)e^{1-t^2}[/tex].
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Write the fraction 16/28 in simplest form.
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Answer:
1/4
Step-by-step explanation:
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The width of a rectangle is 3.8 cm correct to 1 decimal place.
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Answer:
38.75
Step-by-step explanation:
perimeter = 2(15.6+3.8) =38.8
lower bound = the smallest value that would round up to the estimated value =38.75
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Find two points points on x-axis which are at a distance of 5 units from (5,5).
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Step-by-step explanation:
Mark buys last year's best-selling novel, in hardcover, for $23.20. This is with a 20% discount from the original price. What was the original price of the novel?
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Answer:
[tex]29[/tex]
Step-by-step explanation:
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( 1 - 20% ) x = 23.2
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[tex] x = 29[/tex]
Reduce 6x /12x is it 1/2
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Answer:
tero bau ko tauko got it sjdhhrurjrbr
Answer:
LHS=6x/12x=6/12=1/2
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68 rounded to the nearest hundred
Answers
Answer:
100
Step-by-step explanation:
anything lower then 4 stays anythiing 5 or higher is rounded
so this also works with 40 50 400 500 4000 5000, you gte the point
oodluck on your test or homework! :)
Find the root(s) or zero(s) of the function.
Answers
Answer:
-1/2 , 3
Step-by-step explanation:
In order to find the roots of the function 2x² - 5x - 3
We have to solve the quadratic equation :
2x² - 5x - 3 = 0
Calculating the discriminant Δ :
Δ = b² - 4ac
= (-5)² - 4(2)(-3)
= 25 + 24
= 49
Then
√Δ = √49 = 7
Determining the Zeros :
[tex]x=\frac{-b+\sqrt{b^{2}-4ac} }{2a} = \frac{5+7} {2 \times (2)} =\frac{12}{4} =3[/tex]
or
[tex]x=\frac{-b-\sqrt{b^{2}-4ac} }{2a} = \frac{5-7} {2 \times (2)} =\frac{-2}{4} =\frac{-1}{2}[/tex]
Find the perimeter of this figure.
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12 m
18 m
6 m
[? ] = meters
Answers
Answer:
The perimeter of it is going to be 50
Step-by-step explanation:
This is because when finding the perimeter you add up all the sides so
14 + 12 + 18 + 6 = 50
Answer:
50 m
Step-by-step explanation:
Perimeter
= Sum of the length of all sides
= 14 m + 12 m + 6 m + 18 m
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Answers
I'll get 32 jelly beans
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My sister's fraction = 3/(4+3) = 3/7
The number of jelly beans that I'll get will then be:
= My fraction × Number of jelly beans
= 4/7 × 56
= 4 × 8
= 32
Therefore, I'll get 32 jelly beans
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(PLEASE EXPLAIN HOW YOU GOT THE ANSWER) Which inequalities are true? Select the four correct answers.
0 less-than StartRoot 8 EndRoot
StartRoot 8 EndRoot less-than 3
StartRoot 8 EndRoot greater-than StartRoot 7 EndRoot
StartRoot 8 EndRoot less-than 2
StartRoot 8 EndRoot greater-than 8
1 less-than StartRoot 8 EndRoot
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9514 1404 393
Answer:
T, T, F, F, T
Step-by-step explanation:
First of all, it is helpful to know ...
√8 ≈ 2.8284
√7 ≈ 2.6458
Now you can check the inequalities:
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3 and 8 are right of √8 on a number line, so are greater than √8.
Answer:
T, T, F, F, T.
Step-by-step explanation:
Person above is correct!!
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Answer:
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Step-by-step explanation:
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Answers
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54 is 3/5 of 90, therefore Johannesburg's sister is 54cm tall
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Answer:
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Step-by-step explanation:
3- 22
Choose the expressions which are equivalent to the given expression
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B. 3- (+22)
C. 3 + (-22)
D. 22 - 3
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H. 22 +(-3)
Answers
Answer:
» B. 3- (+22) and C. 3 + (-22)
» A. 3- (-22) and E. 22 + 3
» D. 22 - 3 and H. 22 +(-3)
Step-by-step explanation:
[tex].[/tex]
Answer:
3 - (+22)
3 + (-22)
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Step-by-step explanation:
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Answer:
-5,-4,-3,-2,-1,0,1,2,3,4,5
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Answers
Answer:
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Step-by-step explanation:
60 pounds of seed = 7 acres
We want to find acres per pound of seed, so we want to make one side of the equation equal to 1 pound of seed. To do this, we can divide both sides by 60, as anything divided by itself equals 1. We thus have
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Step-by-step explanation:
No of Student in Debre Berhan University - 80 students
No of student good in mathematics - 45 student'sNo of students good in mathematics and English -15
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~V
= sin(x) cos(y) tan(z) i + x^2y^2z^2 j + x^4y^4z^4 k
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Given
[tex]\vec v = f(x,y,z)\,\vec\imath+g(x,y,z)\,\vec\jmath+h(x,y,z)\,\vec k \\\\ \vec v = \sin(x)\cos(y)\tan(z)\,\vec\imath + x^2y^2z^2\,\vec\jmath+x^4y^4z^4\,\vec k[/tex]
the curl of [tex]\vec v[/tex] is
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Please help What's the volume of a cube with a side length of 9 cm? Question 19 options: 701 cm³ 729 cm³ 697 cm³ 752 cm³
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b. ¿Qué fracción representa el total de sillas ocupadas durante los tres días?
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The fraction that represents the total number of seats occupied during the three days is 1.
The total number of seats in the auditorium is 200.
On day 1, 40% of the seats were occupied. So, 40/100 * 200 = 80 seats were occupied on day 1.
On day 2, 35% of the seats were occupied. So, 35/100 * 200 = 70 seats were occupied on day 2.
On day 3, 25% of the seats were occupied. So, 25/100 * 200 = 50 seats were occupied on day 3.
The total number of seats occupied during the three days is 80 + 70 + 50 = 200 seats.
The fraction that represents the total number of seats occupied during the three days is 200/200 = 1.
Here is the solution in mathematical form:
Let x be the fraction that represents the total number of seats occupied during the three days.
Then, x * 200 = 80 + 70 + 50
x = (80 + 70 + 50) / 200
x = 1
Therefore, the answer is 1.
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Answer:
[tex]take \: the \: number = x \\ thenx \times 6 = 2 \times 19 \\ \\ 6x = 38 \\ x = \frac{38}{6} \\ x = 6.33 \\ thank \: you[/tex]
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Which three of the paper strips can be used to form a triangle?
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Step-by-step explanation:
1. 3cm, 5cm, 7cm
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Answers
9514 1404 393
Answer:
76
Step-by-step explanation:
For some common difference d, the terms of the progression are ...
k = 3 +d
w = k +d = 3 +2d
18 = w +d = 3 +3d
Solving for d, we get ...
15 = 3d . . . . subtract 3
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3k +4w = 3(3 +d) +4(3 +2d) = 9 +3d +12 +8d = 21 +11d
3k +4w = 21 +11(5) . . . . substitute for d
3k +4w = 76
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9514 1404 393
Answer:
a) see attached
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c) 201, the number of tiles is 2n+1 for figure n.
Step-by-step explanation:
a) The attachment shows the next two figures
__
b) The pattern adds one more tile at the end of each leg than the previous figure had. This rule matches the figures shown.
__
c) The rule found in part B means that Figure N will have 2N+1 tiles, the N tiles in each leg plus the one where the legs meet.
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Select all of the answers below that are equivalent to G = {Nelson, Niven, Lazenby, Dalton, Moore, Brosnan, Connery, Craig}
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Answer:
Step-by-step explanation:
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Please answer this question
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Step-by-step explanation:
Using the properties of logarithms, the left side of the equation becomes
[tex]\log{3x^3} - \log{x^2} = \log{3x}[/tex]
while the right hand side becomes
[tex]\log{27} - \log{x} = \log{\frac{27}{x}}[/tex]
so we end up with
[tex]3x^2 =27 \Rightarrow x = 3[/tex]