When do polynomials eventually differentiate to zero?

Question

Suppose I have a generic polynomial:

$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x+a_0$$

If I continually differentiate $f(x)$, when will it end up as $f^{(z)}(x)=0$?

For example, given $f(x)=16x^3-x^2-3x+10$:

$$f'(x)=48x^2-2x-3\\f''(x)=96x-2\\f'''(x)=96\\f''''(x)=\color{red}{0}$$

It ends up as $f^{(4)}=0$.

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I know that for the sine function this is not the case:

$$f(x)=\sin(x)\\f'(x)=\cos(x)\\f''(x)=-\sin(x)\\f'''(x)=-\cos(x)\\f''''(x)=\sin(x)\\\text{loop!}$$

Edit: This isn't a polynomial. $x$ is not in the base, among other things.

And if I define

$$f(x)=\sin(x)=\frac{e^{-ix}}{2}+\frac{e^{ix}}{2}$$

Edit: Then we have sine still not as a polynomial. When differentiating it, we get:

$$f'(x)=\frac{1}{2}ie^{ix}\left(-1+e^{2ix}\right)\\f''(x)=\frac{1}{2}e^{-ix}\left(1+e^{2ix}\right)\\f'''(x)=-\frac{1}{2}ie^{-ix}\left(-1+e^{2ix}\right)\\f''''(x)=\frac{e^{-ix}}{2}+\frac{e^{ix}}{2}\\\text{loop 2.0}!$$

What is "special" about the sine function that it does not eventually differentiate to $0$?

As I investigated this question, I started with $i$ as a possible culprit:

$$f(x)=10ix\\f'(x)=10i\\f''(x)=0\\\text{nope}$$

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$$f(x)=10x^i\\f'(x)=10ix^{-1+i}\\f''(x)=(-10-10i)x^{-2+i}\\f^{(13)}(x)=(2716272000 - 8395946000i)x^{-13+i}\\\text{aha!}$$

I found that using $i$ in the exponent led to a polynomial that did not eventually differentiate to $0$.

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When do polynomials eventually differentiate to $f^{(z)}(x)=0$? Is using $i$ in the exponent the only case where it does not?

Answer 1

A finite number of derivatives get to constant zero if and only if the original really is a polynomial. For degree $n,$ the $n+1$ derivative gives zero.

This is also the test, given a sequence of integers, for detecting whether it is given by a polynomial; this is the very simplest application of "finite differences." If I begin with $$ 1, \; 8, \; 27, \; 64, \; 125, \; 216, \; 343, $$ first difference sequence $$ 7, \; 19, \; 37, \; 61, \; 91, \; 127, $$ second differences $$ 12, \; 18, \; 24, \; 30, \; 36, $$ third $$ 6, 6,6,6,6, $$ fourth $$ 0,0,0,0 $$

Answer 2

Think of it the other way round: start from $0$, integrate it $n$ times... you'll end up with a polynomial of degree $n-1$. This means that if a function becomes $0$ (the function, not the number) after a finite number of differentiations, it can only be a polynomial.

Answer 3

We can represent $\sin$ as sort of a polynomial, except with infinitely many terms, namely by power series expansion $$ \sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}=\sin x $$ which perhaps you have seen. What happens when you apply $$ \frac{d^k}{dx^k}\sin(x) $$ for any finite number $k$?

Answer 4

This is a consequence of the nature of "infinity". Sin/cos indeed seem to be polynomials , but of infinite order. So when you differentiate them, however often, the highest order term is still nonzero of infinite order. Therefore you never get to zero value as this would require a zero highest order term (and all other terms).

In fact (see comment below) once a 'polynomial' has an infinite number of terms, it isn't considered to be a polynomial. It is formally considered to be 'power series' instead. The introduction of infinitely many terms gives them a whole bunch of different properties (potentially) that polynomials don't have. As simple examples, polynomials are always well defined, smooth, continuous, non-periodic, even if they have a very large number of terms. But once you extend a polynomial to allow an infinite number of terms (= a power series) like those in the OP, or complex powers, or powers not in the natural numbers (0,1,2...), none of these properties of polynomials are guaranteed to still be the case.

You can't get round this by trying "infinite differentiation" either, as has been suggested. A simple reason is that an infinite set less an infinite set (each differentiation reduces the list of {ax^n} by one) can still leave an infinite set. Or almost any other size set of similar/lower cardinality (it's not always well defined and set theory isn't my strong point but you get the idea). Once you have infinitely many terms, then however much you differentiate, you can still have infinitely many terms.

Answer 5

Somewhat tongue-in-cheek answer: differentiating a function n times will result in a function identically equal to zero iff after n terms in the Taylor series, all the terms are zero. Which is just a roundabout way of saying "all the terms in the polynomial have exponent less than n". Sine's Taylor series goes on forever, so it derivatives go on forever.