# Vocabulary/pdotdot

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`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.

### More Information

1. Boolean x and y are always promoted to floating-point.

### Related Primitives

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