1) Find the slope of the tangent line to the curve 6sin(x)+6cos(y)−4sin(x)cos(y)+x=3π at the point (3π,3π/2).?

6 sin(x) + 6 cos(y) - 4 sin(x)cos(y) + x = 3π

Take differentiation on the both sides :
(d/dx)[6 sin(x) + 6 cos(y) - 4 sin(x) cos(y) + x] = (d/dx)(3π)
6 cos(x) - 6 sin(y) (dy/dx) - 4 sin(x) [-sin(y)] (dy/dx) - 4 cos(y) cos(x) + 1 = 0
6 cos(x) - 6 sin(y) (dy/dx) + 4 sin(x) sin(y) (dy/dx) - 4 cos(x) cos(y) + 1 = 0
6 sin(y) (dy/dx) - 4 sin(x) sin(y) (dy/dx) = 6 cos(x) - 4 cos(x) cos(y) + 1
[6 sin(y) - 4 sin(x) sin(y)] (dy/dx) = 6 cos(x) - 4 cos(x) cos(y) + 1
dy/dx = [6 cos(x) - 4 cos(x) cos(y) + 1] / [6 sin(y) - 4 sin(x) sin(y)]

Slope at the point (3π, 3π/2)
= dy/dx when (x = 3π) and (y = 3π/2)
= [6 cos(3π) - 4 cos(3π) cos(3π/2) + 1] / [6 sin(3π/2) - 4 sin(3π) sin(3π/2)]
= 5/6

6cosx-6siny dy/dx -4[cosycosx-sinxsiny dy/dx] = -1 at (3π,3π/2)
-6+6dy/dx-4[0-0dy/dx] = -1
dy/dx = 5/6