We shall by computation discover that the polynomial functions
for , where
- (multiplication n times),
appear as solutions of the sequence of initial value problems for
- (same as ) for ,
where denotes differentiation with respect to t and . We shall thus discover that
To check, we compute successively for and verify by observing that blue and black curves agree as well the red line and black lines.
We have thus found by time stepping that the derivatives of the monomial polynomials (n times) are given by
- for .
Summary: We have constructed the monomials as solutions to the differential equation . Or the other way around, we have shown that the derivative of is equal to .
We have done this for and can extend to by time stepping backwards in time t. We can also let t represent a spatial variable x and write
- for .
We can finally extend to with , with a special rule for to be discovered as .
1. Experiment with different time steps dt.
2. To see that if , compute formally
to find by dividing by dt
if dt is vanishingly small (with a difference of size dt else). Of course also consider the case n=1 to (show that df/dt = 1 if f(t) = t).
3. Do the same computation for and s0 on.
4. Compute the derivative of a polynomial with coefficients (numbers) .