# A simple question about 1D finite element derivatives

For 1D derivative we have $$F(x) = \frac{\partial f(x)}{\partial x}$$ $$f(x)=\sum_{i}f_ie_i(x)$$ $$F(x)=\sum_{i}F_ie_i(x)$$ where $e_i(x)$ are the FEM basis functions. We can then apply the Galerkin procedure to the derivative,and then get the matrix form $$A\widehat{F}=B\widehat{f}$$ $$\widehat{F}=A^{-1}B\widehat{f}$$ $\widehat{F}$and$\widehat{f}$are the vector form of $F_i$and $f_i$.

For the derivative of the product of two functions $f(x)$and $g(x)$,applying product rule we get $$\frac{\partial (f(x).g(x))}{\partial x}=f(x). \frac{\partial g(x)}{\partial x}+g(x). \frac{\partial f(x)}{\partial x}$$

Now in FEM I want the derivative matrix $C=A^{-1}B$ also to fulfill the similar constraint $$C \widehat{fg} = [f]_{dig}.C\widehat{g} + [g]_{dig}.C\widehat{f}$$ where $[f]_{dig}$ means diagonal matrix with the values $f_i$ in it.

Does any body know what kinds of basis function $e_i(x)$ I can chose?