Find both ∂z/∂u and ∂z/∂v , given that F (x, y, z) = xy^2 + z^3 + sin(xyz) = 0.

Solve for ∂w/∂y and ∂w/∂x w=(1-2y)(x2-xy)

Find the directional derivative of ø = x² + y² + z² at point (1, 2, 1) for a direction determined by dx = 2dy = -2dz.

Find (∂z)/(∂s) and (∂z)/(∂t), given: (d) z = sin (4x + 5y); x = s + t, y = s - t

Let r(t) =< 3t, 5t^2, 2t >. Find the unit tangent vector T(t).

Find ∂f/∂x and ∂f/∂y.

Find the directional derivative of f(x, y, z) Prev x'y at the point (-5, 1, 1) in the direction of the vector v = (1, -3, 4).

Show that (∂z/∂x)2 + (∂z/∂y)2 = (∂z/∂r)2 + (1/r2) (∂z/∂θ)2

a) Find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, 1) in the direction 2i - j - 2k.

Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y) 3e* sin(y), (0, x/3), v = (-5, 12) DyF(0, x/3) =