1.(1)

using TySymbolicMath
@variables x y z
f = 2 * x * y * tan(x) + exp(3 * x * z) + cos(y * z)
a = derivative(g,x)
b = derivative(g,y)
c = derivative(g,z)

1.(2)

@variables x y z
g = (x + y + y * z) / sin(x * y)
a = derivative(g,x)
b = derivative(g,y)
c = derivative(g,z)

2.(1)

x, y = meshgrid2(-1:0.001:1, -1:0.001:1)
T = @. 100 * exp(-x^2 - 2 * y^2)

Tx, Ty = TyMath.gradient(T, 0.001)

indt = find(@. (abs(x - 0.5) < 0.001 && abs(y + 0.5) < 0.001))
grad = [Tx[indt] Ty[indt]]

Tx1(x, y) = 100 * exp(-x^2 - 2 * y^2) * (-2 * x)
Ty1(x, y) = 100 * exp(-x^2 - 2 * y^2) * (-4 * y)
grad = [Tx1(0.5, -0.5) Ty1(0.5, -0.5)]

2.(2)

dx = 3
dy = 4
u = [dx, dy] / norm([dx, dy])
d = dot(grad, u)