A `list`

could be used as a simple representation of a polynomial, $P(x)$, with the items as the coefficients of the successive powers of $x$, and their indexes as the powers themselves. Thus, the polynomial $P(x) = 4 + 5x + 2x^3$ would be represented by the list `[4, 5, 0, 2]`

. Why does the following attempt to differentiate a polynomial fail to produce the correct answer?

```
>>> P = [4, 5, 0, 2]
>>> dPdx = []
>>> for i, c in enumerate(P[1:]):
... dPdx.append(i*c)
>>> dPdx
[0, 0, 4] # wrong!
```

How can this code be fixed?

The problem is that `enumerate`

, by default, returns the indexes and items of the array passed to it with the indexes starting at 0. The array passed to it is the slice `P[1:] = [5, 0, 2]`

and so `enumerate`

generates, in turn, the tuples `(0, 5)`

, `(1, 0)`

and `(2, 2)`

. However, for our derivative we need the indexes into the original list, `P`

, giving `(1, 5)`

, `(2, 0)`

and `(3, 2)`

. There are two alternatives: pass the optional argument `start=1`

to `enumerate`

or add 1 to the default index:

```
>>> P = [4, 5, 0, 2]
>>> dPdx = []
>>> for i, c in enumerate(P[1:], start=1):
... dPdx.append(i*c)
>>> dPdx
[5, 0, 6]
>>> P = [4, 5, 0, 2]
>>> dPdx = []
>>> for i, c in enumerate(P[1:]):
... dPdx.append((i+1)*c)
>>> dPdx
[5, 0, 6]
```