Algebraic approximation of the first derivative

Let us consider a function \(u(t)\) that depends continously on \(t\). The derivative of \(u(t)\) is:

(1)\[ \frac{du}{dt} = \lim_{\Delta t \to 0} \frac{u(t+\Delta t)-u(t)}{\Delta t}. \]

We can obtain an algebraic approximation to \(du/dt\) simply by relaxing the requirement that \(\Delta t \to 0\), retaining a finite \(\Delta t\):

(2)\[ \frac{du}{dt} \approx \frac{u(t+\Delta t)-u(t)}{\Delta t}. \]

If instead of a continuous \(t\), we use discrete values \(t^n=n\Delta t\), we can write the algebraic approximation (2) as:

(3)\[ \frac{du}{dt} \approx \frac{u^{n+1}-u^n}{\Delta t}, \]

where \(u^{n+1}=u(t^{n+1})=u(t^n+\Delta t)\) and \(u^{n}=u(t^n)\).

When dropping the requirement \(\Delta t \to 0\) to obtain the algebraic approximation, we are introducing errors in the calculation of \(du/dt\). We must now find out how large can these errors be.