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].
Learn more about Chain Rule:
https://brainly.com/question/2285262
Related Questions
please solve this question i need help
Answers
Answer:
a=50
b=130
c=130
Step-by-step explanation:
2a-50=a(vertically opposite angles)
a=50
a+c=180(angles on straight line)
50+c=180
c=180-50=130
b=c(vertically opposite angles)
b=130
A number is multiplied by 8, and that product is added to 2. The sum is equal to the product of 2 and 21. Find the number
Answers
Answer:
The number is 5
Step-by-step explanation:
A number is multiplied by 8 = 8x
8x + 2
8x + 2 = 42 Subtract 2 from both sides
- 2 -2
8x = 40 Divide by 8
x = 40/8
x = 5
log base 4 (x+3)²- log base 4 y² = 0
Answers
Answer:
x = - |y| - 3
Step-by-step explanation:
What is the value of (-1/4 - 1/2) ÷ (-4/7)
Answers
Answer:
1 5/16
Step-by-step explanation:
(-1/4 - 1/2) ÷ (-4/7)
Parentheses first
Get a common denominator
(-1/4 - 1/2 *2/2)
(-1/4 - 2/4)
-3/4
Replace in the equation
(-3/4 ) ÷ (-4/7)
Copy dot flip
-3/4 * -7/4
21/16
1 5/16
[tex]\\ \rm\Rrightarrow \left(\dfrac{-1}{4}-\dfrac{1}{2}\right)\div \dfrac{-4}{7}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{-1-2}{4}\times\dfrac{-7}{4}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{-3}{4}\times \dfrac{-7}{4}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{21}{16}[/tex]
[tex]\\ \rm\Rrightarrow 1\dfrac{5}{16}[/tex]
The product of 9 and 4 is divided by 6 .Make mathematical expressions and simplify
Answers
Answer:
The answer is 6
Step-by-step explanation:
9×4/6
=36/6
=6.
the inequality -3n<12 is equivalent to
a. n < 4
b. n < -4
c. n > 4
d. n > -4
e. n > 36
Answers
Answer: D. n > -4
Step-by-step explanation:
Given
-3n < 12
Divide both sides by -3. REMEMBER to change the [>,<] signs when multiplying or dividing a negative term
-3n / -3 > 12 / -3
n > -4
Hope this helps!! :)
Please let me know if you have any questions
Hey there! I'm happy to help!
Inequalities are solved like normal equations with a few extra rules.
So, we could simply divide both sides by -3 to get our answer, but n<-4 would actually be incorrect. With inequalities, dividing by a negative number means you have to flip the inequality sign. This is because you turn the other side negative so you need to adjust the inequality sign to match it.
So, our final answer will actually be n>-4.
Have a wonderful day and keep on learning! :D
5x + 4m = 43my + 6 solve for m
Answers
9514 1404 393
Answer:
m = (5x -6)/(43y -4)
Step-by-step explanation:
Subtract 4m+6 to separate m-terms from other terms.
5x +4m -(4m +6) = 43my +6 -(4m +6)
5x -6 = 43my -4m
5x -6 = m(43y -4) . . . . factor out m
(5x -6)/(43y -4) = m . . . . divide by the coefficient of m
Solve 4/5x + 1/3 = 3
Answers
The answer is 10/3.
In the Cartesian coordinate system, which coordinate point represents the location of the
origin?
A. (0, 1)
B.(0, 0)
C.(1.O)
D.(1, 1)
Answers
Answer:
(0, 0)
Step-by-step explanation:
Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0).
A 260-inch pipe is cut into two pieces. One piece is four times the length of the other. Find the lengths of the two pieces.
The short piece is inches long.
Answers
Ok bye
4w-41 = -7(w-2) Simplify your answer as much as possible.
Answers
Answer:
w = -13
Step-by-step explanation:
4w - 41 = 7(w-2)
Open the bracket
4w -41 = 7w -14
Collect like terms
4w - 7w = 41 - 14
-3w = 27.
Divide both side by -3
w = -9
Answer: w = 5
Step-by-step explanation:
4w-41=-7w+14
4w+7w=14+41
11w=55
W = 55/11
W = 5
[tex]\frac{1}{\sqrt{x} 5}[/tex]
Answers
Answer:
[tex] \frac{ \sqrt{5x} }{5x} [/tex]
Step-by-step explanation:
[tex] \frac{1}{ \sqrt{x5} } [/tex]
[tex] = \frac{1}{ \sqrt{5x} } [/tex]
[tex] = \frac{1}{ \sqrt{5x} } \times \frac{ \sqrt{5x} }{ \sqrt{5x} } [/tex]
[tex] = \frac{1 \sqrt{5x} }{ \sqrt{5x \sqrt{5x} } } [/tex]
[tex] = \frac{\sqrt{5x} }{ \sqrt{5x \sqrt{5x} } } [/tex]
[tex] = \frac{ \sqrt{5x} }{5x} [/tex]
I need to make n the subject
Answers
hope it's helpful
THANK YOU
Add or Subtract: show all work 30 POINTS!!
Answers
Hope this helps! (I think I did my math right…)
Work:
X+2+3+4= x+9
X^2+x+x= x^4
-1+-1+1= -1
(3,-8) and (7,-2)
Jeejekrjrjeksksndnfkfk
Answers
Answer:
First question(-14)
Second question(-24)
Step-by-step explanation:
8 x 3 = 24/since the signs are different the number will be negative.
7 x -2 = -14/since the signs are different the number will be negative.
hope this helps!! plz mark as brainly if it helps you<3
When 3 is added to a number, and the result is multiplied by 5, you obtain two times the number
Answers
Hi
to solve you must transcrive every word in math terms correctly.
so let's first give a name to our mysterious number. As you know, in math we generally call unknown by letter. " X"
so let's now play the detective.
what do we know on " X" ?
we add 3 to X so we have : X+3
then we multiply by 5. so. ( X+3) *5 .
if we do that we have twice X , so 2X
so ( X+3) * 5 = 2 X
now that we have translatted our data in math term, let's unmask. " X"
( X+3) *5 = 2X
5*X +3*5 = 2X
5X +15 = 2X
5X-2X = -15
3X = -15
X = -15/3
X = -5
So is " -5" is our guy.
let's check it out to be sure .
-5 +3 = -2
-2*5 = -10
-10 is twice -5. so -5 is the correct answer.
Ps : some operation are commonly implied. if you do not. know how I could write some line in the calculus, dont hesitate to ask in comments.
There is no mysteries in math everything is done according to rules and can be explained. So don't hesitate !
(5pg)3=125pg. Is wrong or correct
Answers
Answer:
125 p^3 g^3
Step-by-step explanation:
(5pg)^3
We know that (ab)^c = a^c * b^c
5^3 p^3 g^3
125 p^3 g^3
A building is 1 foot from a 7 foot fence that surrounds the property a worker wants to wash a window in the building 10 foot from the ground he plans to place a ladder over the fence so it rest against a building he decides he should place a ladder 7 feet from the fence for stability to the nearest 10th of a foot how long with the latter need to be
Answers
Answer:
The guitar's volume was amplified, but it was hard to hear the notes since it was so loud.
What word or phrase is a signal word that a context clue might be nearby to help determine the meaning of "amplified"?
Step-by-step explanation:
enter the range of values for x:
Answers
12x because you multiply it
What is 7x-4(3x-2)=18
Answers
i got this answer from photo math, you should download it. it give you all the answers and even explains how they got the answer!!
When solving secθ=−4.2 for 180°≤θ≤360°, how many possible answers are there?
0
1
2
unable to determine
Answers
8x – 3 = 11 + 6x
What is x
Answers
Answer:
x = 7
Step-by-step explanation:
[tex]8x-3=11+6x\\\\2x-3=11\\\\2x=14\\\\x=7[/tex]
Answer:
x = 7
Step-by-step explanation:
8x - 3 = 11 + 6x
8x - 3 + 3 = 11 + 6x +3
8x = 6x + 14
8x - 6x = 6x + 14 - 6x
2x = 14
2x/2 = 14/2
x = 7
4x-2=18x+5
Show ur work
Answers
Step-by-step explanation:
[tex]4x - 2 = 18x + 5[/tex]
Collect like terms and simplify
[tex] - 2 - 5 = 18x - 4x \\ - 7 = 14x[/tex]
Divide both sides of the equation by 14
[tex] \frac{ - 7}{14} = \frac{14x}{14} \\ - \frac{1}{2} = x[/tex]
Answer:
x = -1/2
Step-by-step explanation:
4x-2=18x+5
-2-5=18x-4x
-7=14x
-7/14=14/14x
-1/2=x
Finds real root off(x)=x3 +x2 +x+7=0
Answers
9514 1404 393
Answer:
x ≈ −2.10487278573
Step-by-step explanation:
I find a graphing calculator to be a very useful tool for the purpose. It shows the only real root is irrational and gives an approximation as x=-2.105.
The calculator also provides the means to perform Newton's method iteration easily to refine the value of the root to full calculator precision.
__
The Newton's method iterator for finding the next iteration of the root (x') is ...
x' = x - f(x)/f'(x)
Starting with the approximation shown on the graph, it only takes two iterations to get the root to the full precision of the calculator.
write 0.8 as a fraction in simplest form.
Answers
The answer would be 8/10, but the simplified answer would be 4/5. I hope this helped answer your question for you. Enjoy your day, and take care!
a van is traveling 64 7/8 miles per hour. an airplane is traveling 5 6/7 times faster. about how fast is the airplane traveling? could u please help?
Answers
Answer:
it will be 736 speed faster than
the value of the power 4⁴=
Answers
Answer:
256
Step-by-step explanation:
4⁴
= 4 × 4 × 4 × 4
= 16 × 16
= 256
Eunice buys dog bed with a listed price of $31. Eunice uses a sale that provides 30% off. What is the total
final price Eunice pays?
A. $21.70
B. $23.70
C. $25.55
D. $28.10
Answers
Answer:
21.70
Step-by-step explanation:
First find the discount
31 * 30%
31 *.30
9.3
Subtract from the original amount
31 - 9.3
21.70
Help me pleaseeeeeeeee
||x-2|-6|=8
Answers
Answer:
x = -12 or x = 16
Step-by-step explanation:
| x - 2 | - 6 =8
+6 +6
------------------------
| x - 2 | = 14
---> we get x = -14
---> we get x = 14
so, x = -12, or, x = 16
The price of a sandwich decreases from $6 to $4. What is the percent decrease in the price of the sandwich?
Answers
Answer:
33.33%
Step-by-step explanation:
Decrease in price= Original price -- New price
= $6 - $4
= $2
Percentage decrease= $2÷$6 × 100
= 33.33%
