I'm doing some basic calculus exercises on higher derivatives. But I'm stuck at a problem. The question is to find the 1469th derivative of $f(x)=x^{532}-5x^{37}-4$. I've read something about using the general Leibniz rule and a binomial but I can't get my head around it.

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    $\begingroup$ Maybe easier: what is the 533'rd derivative? $\endgroup$ Jul 2, 2015 at 10:32
  • $\begingroup$ It might be good for you to know the existence of Faá di Bruno's generalization of the chain rule to higher derivatives. $\endgroup$
    – Red Banana
    Jul 2, 2015 at 10:47
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    $\begingroup$ @Voyska You don't even need the ordinary chain rule to differentiate a univariate polynomial. $\endgroup$ Jul 2, 2015 at 21:49
  • $\begingroup$ @DavidRicherby I know. I just said it to enhance his mathematical culture. $\endgroup$
    – Red Banana
    Jul 3, 2015 at 0:31
  • $\begingroup$ Do you know what the degree of a (non-zero) polynomial is? Have you ever thought about what happens to the degree of a polynomial when you take the derivative? $\endgroup$ Jul 3, 2015 at 12:58

2 Answers 2


You don't need Leibniz's rule.

Hint : $(x^{532})'=532x^{531},(x^{532})''=(532\times 531)x^{530},(x^{532})'''=(532\times 531\times 530)x^{529}$, etc.

So, what happens when you hit the $533$th derivative ?

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    $\begingroup$ Yes I noticed it, thanks for the quick response! $\endgroup$
    – John
    Jul 2, 2015 at 10:34

Hint: $(x^n)^{(k)}=0$ if $k>n$, $k,n\in\mathbb{Z}_+$.

  • $\begingroup$ This may help OP to find the answer to the question, but it doesn't help anyone since the entirety of the logic behind the solution is skipped in the mad rush to reach the end. The point of this question is to find this result, not to start with it and then trivially write down the answer. $\endgroup$
    – SimonT
    Jul 2, 2015 at 18:52
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    $\begingroup$ I think this answer is fine, given the simplicity of the question: it clearly and concisely expresses the key principle involved, period. $\endgroup$ Jul 4, 2015 at 6:39

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