You want to derive the following equation:
W(x) = A * sin(w * (D.x * x + D.y * z) + t * phi)
= A * sin(w * D.x * x + w * D.y * z + t * phi)
which is the above formula with the expanded dot product. Because we want to find the derivative with respect to x, all other variables (except x) are considered constant. So we can substitute the constants:
c1 = A
c2 = w * D.x
c3 = w * D.y * z + t * phi
W(x) = c1 * sin(c2 * x + c3)
The derivative is:
W'(x) = c1 * c2 * cos(c2 * x + c3)
Reverting the substitution we get:
W'(x) = A * w * D.x * cos(w * D.x * x + w * D.y * z + t * phi)
which describes the y-component of the tangent at a given position.
Similarly, the bitangent (derivative with respect to z) can be described by
W'(z) = A * w * D.y * cos(w * D.y * z + w * D.x * x + t * phi)
Therefore:
tangent = (1, W'(x), 0)
= (1, A * w * D.x * cos(w * D.x * x + w * D.y * z + t * phi), 0)
bitangent = (0, W'(z), 1)
= (0, A * w * D.y * cos(w * D.y * z + w * D.x * x + t * phi), 1)