You can find dy/dx using implicit differentiation. This allows us to solve for dy/dx without needing to first solve for y (sometimes it might even be impossible to solve for y). Simply differentiate each side with respect to x, and use the chain rule whenever you encounter y, recalling that y is some function of x. Doing so gives
d/dx cos(4x-y) = d/dx (x+y)
-sin(4x-y) d/dx(4x-y) = 1 + dy/dx
-sin(4x-y) (4-dy/dx) = 1 + dy/dx
-4sin(4x-y)+sin(4x-y)dy/dx = 1 + dy/dx
sin(4x-y)dy/dx - dy/dx = 1 + 4sin(4x-y)
dy/dx (sin(4x-y) - 1) = 1 + 4sin(4x-y)
dy/dx = (1 + 4sin(4x-y))/(sin(4x-y) - 1).
I hope that helps!