Question

Let f(x,y,z)=xy+z^3, x=r+s−8t, y=3rt, z=s^6.

Use the Chain Rule to calculate the partial derivatives.

(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables

Answer #1

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Let f(x,y,z)=6xy−z^2, x=6rcos(θ), y=cos^2(θ), z=7r.
Use the Chain Rule to calculate the partial derivative.
(Use symbolic notation and fractions where needed. Express the
answer in terms of independent variables

1.
a) Use the Chain Rule to calculate the partial derivatives.
Express the answer in terms of the independent variables.
∂f
∂r
∂f
∂t
; f(x, y, z) = xy +
z2, x = r + s −
2t, y = 6rt, z =
s2
∂f
∂r
=
∂f
∂t
=
b) Use the Chain Rule to calculate the partial derivative.
Express the answer in terms of the independent variables.
∂F
∂y
; F(u, v) =
eu+v, u =
x5, v = 2xy
∂F
∂y
=
c)...

Use the Chain Rule to find the indicated partial
derivatives.
w = xy + yz + zx, x = r
cos(θ), y = r
sin(θ), z = rθ;
∂w
∂r
,
∂w
∂θ
when r = 4, θ =
π
2
∂w
∂r
=
∂w
∂θ
=

Use the Chain Rule to evaluate the partial derivative ∂g/∂u at
the point (u,v)=(0,1), where g(x,y)=x^2−y^2, x=e^3ucos(v),
y=e^3usin(v).
(Use symbolic notation and fractions where needed.)

Use the Chain Rule to find the indicated partial derivatives. u
=sqrt( r^2 + s^2) , r = y + x cos(t), s = x + y sin(t)
∂u ∂x , ∂u ∂y , ∂u ∂t when x = 1, y = 4, t = 0

Use the Chain Rule to find the indicated partial derivatives. u
= x^4 + yz, x = pr sin(θ), y = pr cos(θ), z = p + r; (partial
u)/(partial p), (partial u)/(partial r), (partial u)/(partial
theta) when p = 3, r = 4, θ = 0

Use the Chain Rule to find the indicated partial
derivatives.
z = x2 + xy3,
x = uv2 + w3,
y = u + vew
when u = 2, v = 2, w = 0

Let w(x,y,z) = x^2+y^2+z^2 where x=sin(8t), y=cos(8t) , z=
e^t
Calculate dw/dt by first finding dx/dt, dy/dt, and dz/dt and using
the chain rule
dx/dt =
dy/dt=
dz/dt=
now using the chain rule calculate
dw/dt 0=

Use the Chain Rule to find the indicated partial
derivatives.
u =
r2 + s2
, r = y + x cos
t, s = x + y sin t
∂u
∂x
,
∂u
∂y
,
∂u
∂t
when x = 2, y =
5, t = 0

Let f(x y z)=x4y4+z5,
P=(4, 4, 1).
Calculate the directional derivative in the direction pointing
to the origin. Remember to normalize the direction vector.
(Use symbolic notation and fractions where needed.)
Duf(4, 4, 1)

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