# Vocabulary/pdotdot

 p.. y Polynomial Derivative

Rank 1 -- operates on lists of y, producing a list for each one -- WHY IS THIS IMPORTANT?

The first derivative of a given polynomial y.

got from differentiating the individual terms of the polynomial

Polynomial y can be provided in coefficient form (list of ascending powers), multiplier-and-roots form (m;r), or exponent form (boxed table form). The result is always expressed in coefficient form.

Examples:

 Series 1 y y² y³ y⁴ 1st derivative 1 2y 3y² 4y³
```   p.. 1 1 1 1 1
1 2 3 4
```
 Series 1 2y 3y² 4y³ 5y⁴ 1st derivative 2 6y 12y² 20y³
```   p.. 1 2 3 4 5
2 6 12 20
```

### Common uses

Work with finite approximations of infinite series.

### Related Primitives

Ordinary Derivative (u d. n), Polynomial (x p. y)

 x p.. y Polynomial Integral

Rank 0 1 -- operates on individual atoms of x, and lists of y -- WHY IS THIS IMPORTANT?

The integral of a given polynomial y.

got from integrating the individual terms of the series

The x-argument is the constant of integration that will be added to the result.

Polynomial y can be provided in coefficient form (list of ascending powers), multiplier-and-roots form (m;r), or exponent form (boxed table form). The result is always expressed in coefficient form.

Examples: (invert the examples in monadic p.. above)

 Series 1 2y 3y² 4y³ Integral x y y² y³ y⁴
```   99 p.. 1 2 3 4
99 1 1 1 1
```
 Series 2 6y 12y² 20y³ Integral x 2y 3y² 4y³ 5y⁴
```   1 p.. 2 6 12 20
1 2 3 4 5
```

### Common uses

Work with finite approximations of infinite series.