Two spacecraft are following paths in space given by r, = (sin(t), t, 1² ) and rɔ = (cos(t), 1 – 1, 1³). If the temperature for the points is given by T(x, y, z) = x²y(5 – z), use the Chain Rule to determine the rate of change of the difference D in the temperatures the two spacecraft experience at time t = 3. (Use decimal notation. Give your answer to two decimal places.) dD dt

Q: Find aw/as and ôw/at using the appropriate Chain Rule. Function Point w = y3 - 9x?y x = e, y = e s =…

A: Explained below

Q: Find dy/dx if y = x3/2 by using the Chain Rule with y as a compositionof y = √u and u = x3

A: Recall the chain rule of differentiation:

Q: Use the appropriate form of the chain rule to find the derivatives. w = 11x² + 11y² + z²; x = p sin…

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Q: dz 922 – 4y², x = 2 sec t, y = 3 tan t. Using the chain rule, find dt Given z =

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Q: dz Use an appropriate form of the chain rule to find dt 1 z = 4 cos x – sin xy; y = 7t || Enter the…

A: We have to find

Q: Use the Chain Rule to evaluate the partial derivative at the point specified. ∂f/∂s at (r, s) = (1,…

A: Use the Chain Rule to evaluate the partial derivative at the point specified. ∂f/∂s at (r, s) = (1,…

Q: dw Use the Chain Rule to find where w = cos 12x sin 4y, x = - dt and y = t". 4 dw

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Q: Consider the following. w = cos(x − y),    x = t2,    y = 21 (a) Find dw/dt by using the…

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Q: Use the chain rule to express dy /dx in terms of x and y y = sin t and t = cos x where t=0

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Q: dz 3 Use the Chain Rule to find where z= 2x sin y, x=t°, and y = 3t°. dt '

A: Given,           z=2xsiny , x=t5 and y=3t3.

Q: Ex 3: Use the chain rule to express in terms of x and y dx c) y = ( and x =-1 att 2

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Q: use an appropriate form of the chain rule to find dz/dt. z= 8cosx-sinxy; x=1/t, y=3t

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Q: Find the rate of change of w with respect to / at t = 1 by using the chain rule. You can check your…

A: Given w=x7y3z3     ;    x=t6    , y=t+2    ,   z=3t4

Q: Suppose w = x = cos? t, y = sin? t, z = dw Evaluate at t = 3 , by using the Chain Rule dt

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Q: dz Use an appropriate form of the chain rule to find dt 1 z = 4 cos x – sin xy; x y = 7t t' Enter…

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Q: Find dy/dx if y = x3/2 by using the Chain Rule with y as a compositionof y = u3 and u = √x

A: Given: ,and . Find out by using the chain rule.

Q: dw Use the Chain Rule to find dt w = x²e²y cos(32), x = cos t, y = In (t + 2), z = t

A: The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x).

Q: d use the Chain Rule to calculate f(r(t)). dt f(x, y) = x² – 3xy, r(t)= (cos t, sin t), 1 = }

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Q: For the functions w = 8x + 3y, x= cos t, and y = sin t, express dw as a function of t, both by using…

A: The given function is w=8x2+3y2, x=cost and y=sint. We have to express dwdt as a function of t.…

Q: 3. Let z = x?y + 3xy+ where x = sin 2f and y cos t. Use the chain rule to dz find when t = 0. dt

A: As indicated in the question, we will make use of chain rule of differentiation. Recall that the…

Q: Find aw/as and dw/at using the appropriate Chain Rule. Function Values w = y - 9x3y x = e, y = ef s=…

A: Partial Derivative of Chain Rules

Q: dz Use an appropriate form of the chain rule to find dt z = 7 cos x – sin xy; x = ;, 1 y = 9t Enter…

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Q: Q4) Use the chain rule to express dy /dx in terms of x and y y = sin t and t = cos x where t = 0

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Q: Use appropriate forms of the chain rule to find dz/ðu and dz/dv. Z = cos x sin y; x =u – v, y = u² +…

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Q: Let z = x?y + 3xy+ where x = sin 2t and y dz find = cos t. Use the chain rule to when t = 0. dt

A: See solution below. Using definition of chain rule.

Q: dz Use an appropriate form of the chain rule to find dt 1 z = 8 cos a – sin xy; x = - y = 4t t'…

A: This question is based on Derivative of Function.

Q: Use the appropriate form of the chain rule to find the derivatives. w = 3x? + 3y² + z²; x = p sin ø…

A: Given:   w = 3x2 + 3y2 +z2  x and y and z values is given

Q: 4. Given that c(t) = t² + 2, y(t) = t³ – 2t and f(x,y) = x² sin (xy). Using the df and evaluate it…

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Q: dw Use an appropriate form of the chain rule to find dt 1 w = 8 cos xy – sin cz; x = y=t, z = t7 t'…

A: Here we have to find dw/dt by chain rule.

Q: If = Z sin x = 2t, y = 2 – t2, find dz/dt using the chain rule. Assume the variables are restricted…

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Q: IF V = xsin(y) + ye where x = s + 2t and Y = st Use the chain rule to find dv and- ds dt when s=0…

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Q: Find using the appropriate Chain Rule for W =y- 7x*y where x = e° and y = e", and evaluate the…

A: The given function is w=y3-7x2y where x=es and y=et. Differentiate the function w=y3-7x2y partially…

Q: dx Use the chain rule to find () in terms of x and y dy 2 y = ( and X = t2 at t= 2

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Q: Let T = x²y - xys + 6x, where x = r cos 0 and y = r sin 6. Use the chain rule to find dT/00.

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Q: Let z = x*y + 3xy* where x = sin 2t and y dz find = cos t. Use the chain rule to %3D when t = 0. dt

A: z=x2y+3xy4where x = sin 2t and y = cos t We need to find dzdt when t =0 Let's use the chain rule :…

Q: If Z = /xy + y, x= cos 0 , y= sin 0 when 0 = 2 dz By using chain rule find de

A: According to chain rule of differentiation .According to product rule of differentiation . According…

Q: x²y + 3xy* where x = sin 2t and y = cos t. Use the chain rule to dz find 3. Let z when t = 0. dt

A: To use the chain rule to find dz/dt at t=0

Q: dz Use the Chain Rule to find : z = = cos(x + 4y),x = 5t*, y = - %3D dt

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Q: Use the chain rule to find dz/t. z = x² + y² + xy, x = sint, y = et

A: Here we will find dz/dt,

Q: dy If y = tan t , x = tan? t ,Find by using Chain Rule dx at t =

A: Let the two functions are xt and yt. The chain rule to find dydx is given by the following formula.…

Q: given that, x(t) =t2 +2    ,y(t)=t3 -2t  and f(x,y) =x2 sin(xy) . using the chain rule for partial…

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Q: dw Evaluate at the given value of t, by using the Chain Rule, dt w = 2ye* - Inz, x = ln(t² + 1), y =…

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Q: Suppose w = 2x2 + 8y², Compute dw/dt in two different ways: x = 7 cos(t) + 5 sin(t) , and y = 7…

A: The chain rule is mentioned below: dwdt=∂w∂x·dxdt+∂w∂y·dydt

Q: dz Use an appropriate form of the chain rule to find dt 1 z = 6 cos x – sin xy; Y = 8t %3D t' Enter…

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Q: Find ôw/as and ôw/ôt using the appropriate Chain Rule. Function Values w = sin(7x + 8y) X = s + t, y…

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Q: dz Use an appropriate form of the chain rule to find dt 1 z = 5 cos x – sin xy; y = 4t t' X = Enter…

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Q: Find dy/dx if y = x by using the Chain Rule with y as a compositionof y = 1 + (1/u) and u = 1/(x -…

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Q: dw Use an appropriate form of the chain rule to find dt 1 w = 8 cos xy – sin xz; x = , y=t, z= t'

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Q: dw 3. Given w=ln (x²+y) where x=2t , and y=4-t . Find in two ways: dt (a) Using the appropriate…

A: Given:- w = ln(x ^ 2 + y) where x = 2t , and y = 4 - t . To find:- dw/dt (a) Using the appropriate…

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.