This is probably a very trivial question but I have not been able to figure it out myself, so here goes.
Let $\Gamma$ be a smooth boundary in 2-D divided into $N$ quadratic (3-noded), continuous, finite elements. Consider one such 3-noded element on the boundary $\Gamma$. Let $\phi$ be a continuous function on this boundary. Consider a point $\mathbf{x}$ on $\Gamma$ which resides on one of the finite elements. Now we approximate $\phi(\mathbf{x})$ using the conventional FE shape functions, $$ \phi(\mathbf{x}) = \sum_{j=1}^3 N_j \phi_j $$ where $N_j$ are the polynomial shape functions and $\phi_j$ the nodal values of the function $\phi$ on $\Gamma$. Similarly, we can approximate the normal derivative of the continuous function $\phi$ at point $\mathbf{x}$ on $\Gamma$ as, $$ \frac{\partial \phi(\mathbf{x})}{\partial n(\mathbf{x})} = \sum_{j=1}^3 N_j \frac{\partial \phi_j}{\partial n_j} $$ where $\frac{\partial \phi_j}{\partial n_j}$ is simply the nodal value of the normal derivative of $\phi$ at node $j$. The question is can we alternatively write, $$ \frac{\partial \phi(\mathbf{x})}{\partial n(\mathbf{x})} = \sum_{j=1}^3 \frac{\partial N_j}{\partial n_j}\phi_j $$ where the shape functions are differentiated instead of the function $\phi$? Is it necessary that the normal (or other) derivatives of a function be modeled with linear combination of the shape functions multiplied with the nodal values of the derivatives?