Question

a. Use the Chain Rule to find the indicated partial derivatives.

z = x⁴ + x²y,    x = s + 2t − u,    y = stu²;

∂z

∂s

∂z

∂t

∂z

∂u

when s = 1, t = 2, u = 3

b. 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 = 8, θ = pi/2

c. 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 = 2,  t = 0

d. Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies.

u(4, −5)	=	8		v(4, −5)	=	1
us(4, −5)	=	−2		vs(4, −5)	=	−4
ut(4, −5)	=	−9		vt(4, −5)	=	3
Fu(8, 1)	=	5		Fv(8, 1)	=	0

Find W_s(4, −5) and W_t(4, −5).

e. Use the equations to find

∂z/∂x and ∂z/∂y.

x² + 2y² + 3z² = 1

Answer 1

Solving only first que as per HomeworkLib guidelines

.

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