# What is the $1469^\text{th}$ derivative of $x^{532}-5x^{37}-4$?

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.

• Maybe easier: what is the 533'rd derivative? – David Mitra Jul 2 '15 at 10:32
• It might be good for you to know the existence of Faá di Bruno's generalization of the chain rule to higher derivatives. – Billy Rubina Jul 2 '15 at 10:47
• @Voyska You don't even need the ordinary chain rule to differentiate a univariate polynomial. – David Richerby Jul 2 '15 at 21:49
• @DavidRicherby I know. I just said it to enhance his mathematical culture. – Billy Rubina Jul 3 '15 at 0:31
• 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? – Jeppe Stig Nielsen Jul 3 '15 at 12:58

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