This is a good question.

Chebyshev differential matrices are considered in detail in

M.D. Morgan, Modelling Multi-Fractured, Multi-Well, Tight-Gas Reservoirs, Ph.D. thesis, University of Calgary, Alberta, CA, 2013:

theses.ucalgary.ca/bitstream/.../2/ucalgary_2013_morgan_michael.pdf

See, for example, Section 8.2,  starting on page 125, on Chebyshev spacing, including a detailed formula and implementation for a Chebyshev differential matrix.

Another good place to look is

N. Podoliak, Magneto-optic effects in colloids of ferromagnetic nano particles in nematic liquid crystals, Ph.D. thesis, University of Southhampton, 2012:

http://eprints.soton.ac.uk/338023/1.hasCoversheetVersion/PhD_Thesis_Podoliak_N.pdf

See Appendix A.1, starting on page 149.

Another place to look is

A.C, Densmore, Algorithms for Rapid Characterization and Optimization of Aperture and Reflector Antennas, Ph.D. thesis, University of Califonia, Los Angeles, 2014:

https://escholarship.org/uc/item/4mx9393h#page-6

See Section 6.3.3, starting on page 145.  

This is a good question.

Chebyshev differential matrices are considered in detail in

M.D. Morgan, Modelling Multi-Fractured, Multi-Well, Tight-Gas Reservoirs, Ph.D. thesis, University of Calgary, Alberta, CA, 2013:

theses.ucalgary.ca/bitstream/.../2/ucalgary_2013_morgan_michael.pdf

See, for example, Section 8.2,  starting on page 125, on Chebyshev spacing, including a detailed formula and implementation for a Chebyshev differential matrix.

Another good place to look is

N. Podoliak, Magneto-optic effects in colloids of ferromagnetic nano particles in nematic liquid crystals, Ph.D. thesis, University of Southhampton, 2012:

http://eprints.soton.ac.uk/338023/1.hasCoversheetVersion/PhD_Thesis_Podoliak_N.pdf

See Appendix A.1, starting on page 149.

Another place to look is

A.C, Densmore, Algorithms for Rapid Characterization and Optimization of Aperture and Reflector Antennas, Ph.D. thesis, University of Califonia, Los Angeles, 2014:

https://escholarship.org/uc/item/4mx9393h#page-6

See Section 6.3.3, starting on page 145.  

if you need to second order  Chebyshev' Differential Matrix  for just computational purposes, you might construct the matrix   D  for first order derivative   , after that simply  squaring D, namely D^2, gives you second order derivative matrix.  Depending  on boundary conditions, you need to change first and last columns or rows. 

In trefethen's  book " spectral methods in matlab,  chapter 7 you can see the implementation details.

However if you need to an explicit formula for higher derivatives, you should consult the above mentioned references

just adding the link to the book of  Trefethen mentioned above

http://people.maths.ox.ac.uk/trefethen/book.pdf

Thank you guys, I figured it out :)

For a related method, which may or may not apply to your application, you may want to look at the method in 

"The EPS method: A new method for constructing pseudospectral derivative operators." Sandberg, K. and Wojciechowski, K.J., Journal of Computational Physics, 230(15), p. 5836–5863, (2011)

This method is easy to implement, and may give higher accuracy and (significantly) smaller norms than the traditional constructions.

http://www.sciencedirect.com/science/article/pii/S0021999111002208

Here are the codes for generating the matrix:

http://scicomp.stackexchange.com/questions/24470/how-do-i-form-the-chebyshev-differentiation-matrix-in-matlab

For a more detailed introduction:

http://www.math.nus.edu.sg/~matgkv/Lecture8.pdf