Let w(x, y, z) = x² + y? + z2 where x y = cos(7t) , z = e-6t. sin(4t), %3D %3D dw dx dy dz and using dt Calculate by first finding dt dt dt the chain rule. dx 4 cos (4t) %3D dt dy 7 sin(7t) %3D dt dz %3D dt -be-6t Now use the chain rule to calculate the following: dw %3D dt Hint: Recall the chain rule: dw dw da dy dw dz %3D dt dt dt dz dt of

Q: Use an appropriate form of the chain rule to find dz/dt 1. z = 3x?y; x = t and y = t2 2. z = 3 cos x…

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Q: dw Use an appropriate form of the chain rule to find dt w = 4 cos xy – sin xz; x y = t, z = t8 t'…

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Q: dw Use an appropriate form of the chain rule to find dt 1 w = 6 cos xy – sin xz; x = t' y = t, z =…

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Q: Let w(x, y, 2) = a? + y? + 2? where a = sin( – 9t), y = cos(- 6t), z = e". dw dæ dy dz by first…

A: We have  W= x2+y2+z2  Where , x= sin(-9t) , y= cos(-6t), z= e3t  Find dx/dt, dy/dt, dz/dt and dw/dt

Q: Use the Chain Rule to find dz/ds and dz/dt. z = x°y°, x = s cos(t), y = s sin(t) dz as az at %3D

A: Given:-z=x5y9x=scos⁡t,y=ssin⁡tTo find∂z∂s&∂z∂tBy chain rule∂z∂S=∂z∂x∂x∂s+∂z∂y∂y∂Swe…

Q: dw Use an appropriate form of the chain rule to find dt 1 w = 9 cos xy – sin xz; x = y = t, z = t3…

A: We have to find

Q: Let z(x, y) = x°y where a = t & y = tª. dz Calculate dt chain rule. dx by first finding dt dy and…

A: Given below the detailed solution using chain rule.

Q: Let w(z, y, 2) = x² + y² + z² where z = sin(7t) , y = cos(7t), z = e'. dz dy dw by first finding dt…

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Q: Use the Chain Rule to find dw/dt. + y2 + z2 x = 5 sin(t), y = 7 cos(t), z = 6 tan(t) W = In

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Q: et w(x, y, 2) = x² + y² + 2² where z = sin(7t) , y = cos(6t), z = e". %3D dw dr dy dz by first…

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Q: After using z = X - y as substitution, the DE (x-y)^2 * dy/dx = a^2 (a is constant) transforms to…

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Q: When reduced to a linear DE of order one in z, the Bernoulli's equation dy 2 +y √( 1 ) = 3x²y² =…

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Q: Let z(x, y) = 7e" sin(y) where x = t° & y = 3nt. dz Calculate dt dy and using the chain rule. dt dx…

A: Using chain rule dz/dt =(dz/dx)×(dx/dt)+(dz/dy)×(dy/dt)

Q: Let w(x, y, z) = x² + y² + z² where x = sin(7t), y = cos(-3t), z = e-4t. dx dy dz dw dt dt' dt dt…

A: Given problem:- wx,y,z=x2+y2+z2,x=sin7t,y=cos-3t,z=e-4t Find dwdt

Q: dz Suppose that z = x³ – 3y?, x = 4(t³ – 1), and y = Int. Find using the Chain Rule. dt %3D -

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Q: dw 3) Find dt if w 5 cos(xy) – sin(xy), x =,y = t, z = t3 at at 4) Let t =,u = x² - y? ,v = 4xy³.…

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Q: tan(): '(), Suppose z = x = u cos(5v) , and y = u sin(5v) . Compute dz/du and az/dv in two different…

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Q: Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y =…

A: Given: w=x2e2ycos3z x=cost y=lnt+2 Putting x, y and z in w, we get…

Q: Find dw/dt using the appropriate Chain Rule. Function Value w = x sin y x = e', y = n - t t = 0…

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Q: Let w = x^2e^2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y…

A: Given  By chain rule of differentiation , we have

Q: z = sin x cos y, = VE, y = 1/t x =

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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: Use the Chain Rule to find dz/dt. dz dt || z = √√√ 1 + x² + y², x = 2 In(t), y = cos(t) 1 √1+ (Int)²…

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Q: Let z(x, y) = x'y where x = t° & y = t°. dz by first finding dt dy and using the chain rule dt dx…

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Q: Let w(x, y, z) = x² + y? + z? where x = sin( – 7t) , y = cos( – 8t) , z : dw by first finding dt dx…

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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: Use the Chain Rule to find dz/dt or dw/dt. z=sin x cos y, x= √t, y=1/t

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: Find dw/ dt at t = 1 if w = xey + y sin z - cos z, x = 2sqrt(t), y = t - 1 + ln t, and z = pt.

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Q: Use the Chain Rule to find dw/dt. w = In Vx2 + y2 + z2 x = 2 sin(t), y = 5 cos(t), z = 4 tan(t) dw…

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Q: Use the Chain Rule to find dw/dt. w = In(V? + ° + z2 x = 3 sin(t), y = 2 cos(t), z = 5 tan(t) dw dt

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Q: Let z(x, y) 2x2 – 3y? where a 2s – 4t & y = 9s + 8t. he at dy dx az Calculate ds dz by first finding…

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Q: Given z = 2 cos t +35 and y = 8 sin t %3D %3D dx (i) dt -2*sin(t) %3D dy (ii) 16*sin(t)*cos(t) %3D…

A: Given x=2cost+35 and y=8sin2t

Q: a)Let z = x?y – y² where x = t2 and y = 2t . Find dz/dt at t=1 using chain rule.

A: Given query is to find derivative is nge 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: dy dy dt * dx dt dx Ex3: Use the chain rule to express: in terz dx dy t2 a) y = and t = v2x +1 %3D…

A: Given rule of derivative  dydx=dydt×dtdx

Q: If x'+y' +z-9, dx/dt = 5, and dy/dt = 4, find dz/dt when (x,y,z) (2,2,1)

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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: 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/dt. z = x2 + y2 + xy,    x = sin(t),    y = 3et

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Q: Use the Chain Rule to find dz/dt.z = sqt(8+x^2+y^2,    x = 4 ln t,    y = cos t dz/dt=

A: Please refer to the image below

Q: dw Find using the appropriate Chain Rule for w = cos(4x−4y), where x = tº and y = 3. dt

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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: dw 3) Find ifw = 5 cos(xy) – sin(xy),x = ,y = t,z = t³ - %3D dt

A: Given: w=5cosxy-sinxy, x=1t, y=t, z=t3 For finding dwdt, first we find w in terms of t So, we put…

Q: Use the Chain Rule to find dw/dt. w = In( V x2 + y² + z? ), x = 2 sin(t), y = 7 cos(t), z = 8 tan(t)

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Q: If f(x, y) = 4x²y+7xy°, then the total differential of f at x = 1 and y = 1 is : %3D O df = 15 dx –…

A: Use formula of total differential

Q: Use the Chain Rule to find dw/dt. w = In V x2 + y2 + z2 x = 9 sin(t), y = 3 cos(t), z = 5 tan(t) dw…

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Q: Let w(x, y, z) = x2 + y? + z2 where x = sin( – 2t) , y = cos( – 8t), z = e". 9t dw dx dy dz…

A: To find the derivative using chain rule

Q: Q2) Use the chain Rule to find dw\dt of y w = =+2,x = cos²(t), y = sin² (t), z =, at t = 3

A: In this question,w is function of x,y,z and x,y,z are the function of t so first we find out…

Q: Suppose x, y, and z are positive functions of t. If z x + y?, dx/dt 1, and dy/dt 2, find dz/dt when…

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Q: Use the Chain Rule to find dz/dt. z = sin(x) cos(y),    x = The Square Root of t,    y = 8/t

A: Given: z = sin(x) cos(y),    x = √t,    y = 8/t By Chain Rule,

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.