# Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable?

If so, does it only mean that at some point we reach 0 and then we keep on getting 0?

Thank you!

• That is exactly correct. Good thinking! Jun 29, 2015 at 22:33
• Yes, polynomials are infinitely many times differentiable, and yes, after some finite number of derivatives (specifically $\deg f + 1$) we get $0$, and then we continue to get $0$ thereafter. Jun 29, 2015 at 22:33
• ... but that's not what it means, it's just what happens in this case when you keep differentiating. Jun 29, 2015 at 22:34
• For example, $\sin x$ is infinitely differentiable, but you never reach and stay at zero. Jun 29, 2015 at 22:36

Yes.

$$\frac{d}{dx} 0 = 0,$$ so once you've taken $\deg(p) + 1$ derivatives, where $p$ is the polynomial you're considering, you will just keep getting zero.

• If in this logic, then constant function is also infinitely differentiable? Apr 5, 2017 at 1:40
• Yes --- the constant functions are just a special case of this, being the degree zero polynomials. Apr 10, 2017 at 21:03

The other answers probably answered your question, but in case you wanted to learn more, the key word to look up is smoothness of the function (see here for instance: https://en.wikipedia.org/wiki/Smoothness).

So another, more fancy way of asking your question is whether or not polynomials are smooth functions.

• Only one of the two other answers is above. You may want to change your wording to make it more independent of surroundings. Jun 30, 2015 at 10:28
• Changed the wording. Thanks! Jun 30, 2015 at 13:32