Differentiation of a Lagrange Interpolation Polynomial

The interpolation polynomial defined in Eq. 8.89 is a continuous function and therefore can be differentiated as well as integrated. 

Taking the first and second derivatives of the interpolation polynomial (Eq. 8.89), we obtain 

In most practical problems, only the first two derivatives are required, so these are presented here. However, if higher derivatives are needed, the Lagrange interpolation polynomial can be differentiated further. 

In particular, if we are interested in obtaining the derivative at the interpolation points, we have 

The summation format in the RHS of Eqs. 8.95 and 8.96 suggests the use of a vector representation for compactness, as we will show next. 

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