The Derivative of a Polynomial in Octave

In today's tutorial, we'll explore how to determine the derivative of a polynomial using Octave's polyder(y) function.

polyder(y)

This function requires just one argument: `y`, an array containing the polynomial's numerical coefficients.

Let's get started with a practical example.

Imagine you're working with the polynomial:

$$ P(x) = 2x^3 + 4x + 3 $$

Begin by defining an array with the polynomial's coefficients, organized by degree:

>> P = [2 0 4 3]

To derive the polynomial, input the polyder(P) function:

>> polyder(P)

The output you'll see represents the first derivative:

ans = 6 0 4

This translates to:

$$ P'(x) = \frac{d \ P(x)}{dx} = 6x^2 + 4 $$

For verification purposes, remember that the derivative of a polynomial is the cumulative derivative of its individual terms. By breaking it down:$$ P'(x) = \frac{d \ ( 2x^3 + 4x + 3)}{dx} = \frac{ d \ 2x^3}{dx} + \frac{d \ 4x}{dx} + \frac{d \ 3}{dx} = 6x^2 + 4 + 0 $$

The Second Derivative

To ascertain the polynomial's second derivative, employ the `polyder()` function consecutively:

>> d1=polyder(P);
>> d2=polyder(d1)

>> polyder(polyder(P))

Either method will yield the second derivative:

ans = 12 0

This can be expressed as:

$$ P''(x) = 12x $$

This methodology can be extended to compute higher-order derivatives, be it the third, fourth, or any subsequent derivative of the polynomial.

Thank you for joining this tutorial. With these tools in hand, you're well-equipped to tackle polynomial derivatives in Octave with confidence.