Derive the following discretized equations :

a$.$ First order derivative : $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}{|}_{i,j}$~$|=\frac{u_i^{j+1}-u_i^j}{t}$

b$.$ Second order derivative: $,\frac{de{l}^{2}u}{del{x}^2}{|}_{i,j}$~$|=\frac{u_{i+1}^j-u{T}_i^j+u_{i-1}^j}{(x{)}^2}$