Higher order derivatives: p(x) = (x + x^5 + x^7)^10

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jwpaine

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Let's say I have the function: \(\displaystyle \L p(x) = (x + x^5 + x^7)^{10}\)

And I want to find the nth derivative of p(x).

Now for \(\displaystyle \L f(x) = x^m,\,\, f^{n}(x) = \frac{m!}{(m-n)!}x^{m-n}\)

So if I wanted to find the nth derivative of p(x), could I use the chain rule and sum rule using the factorial method:

\(\displaystyle \L p^{n}(x) = (\frac{10!}{(10-n)!}(x + x^5 + x^7)^{10-n})(\frac{1!}{(1-n)!}x^{1-n}\,+\,\frac{5!}{(5-n)!}x^{5-n}\,+\, \frac{7!}{(7-n)!}x^{7-n})\)

Is this legal?
 
Re: Higher order derivatives

jwpaine said:
Let's say I have the function: \(\displaystyle \L p(x) = (x + x^5 + x^7)^{10}\)

And I want to find the nth derivative of p(x).

Now for \(\displaystyle \L f(x) = x^m,\,\, f^{n}(x) = \frac{m!}{(m-n)!}x^{m-n}\)

So if I wanted to find the nth derivative of p(x), could I use the chain rule and sum rule using the factorial method:

\(\displaystyle \L p^{n}(x) = (\frac{10!}{(10-n)!}(x + x^5 + x^7)^{10-n})(\frac{1!}{(1-n)!}x^{1-n}\,+\,\frac{5!}{(5-n)!}x^{5-n}\,+\, \frac{7!}{(7-n)!}x^{7-n})\)

Is this legal?
Click to expand...

Lets do couple of differentiation:

p'(x) = 10(x + x^5 + x^7)^9 * (1+5x^4+7x^6)

p"(x) = 10*9*(x + x^5 + x^7)^8 * (1+5x^4+7x^6) + 10(x + x^5 + x^7)^9 * (5 * 4 x^3+7*6*x^5)..........No does not work out
 
Then is there a more efficient way of finding the nth derivative than manually doing p'(p'(p'...(x))) ?
 
No, JW, I don't think so. This one gets ickier and ickier.

I ran it through Maple and it gave me, for the nth derivative:

\(\displaystyle \L\ 3630\,{\it pochhammer} \left( 22-n,n \right) {x}^{21-n} + 261280 \,{\it
pochhammer} \left( 46-n,n \right) {x}^{45-n}\)
\(\displaystyle \L + 172140 \,{\it pochhammer} \left( 38-n,n \right) {x}^{37-n} + 211680 \,{\it pochhammer} \left( 40-n,n \right) {x}^{39-n}\)
\(\displaystyle \L + 217550 \,{\it pochhammer} \left( 50-n,n \right) {
x}^{49-n} + 8640 \,{\it pochhammer} \left( 24-n,n \right) {x}^{23-n}\)
\(\displaystyle \L + 810\,{\it pochhammer} \left( 18-n,n \right) {x}^{17-n} + 1800\,{\it
pochhammer} \left( 20-n,n \right) {x}^{19-n}\)
\(\displaystyle \L + 26880\,{\it pochhammer} \left( 28-n,n \right) {x}^{27-n} + 160 \,{\it pochhammer} \left( 16-n,n \right) {x}^{15-n}\)
\(\displaystyle \L + 140\,{\it pochhammer} \left( 14-n,n \right) {x}^{13-n} + 680\,{\it pochhammer} \left( 68-n,n \right) {x}^{67-n}\)
\(\displaystyle \L + 70\,{\it pochhammer} \left( 70-n,n \right) {x}^{69-n} + 240030\,{\it pochhammer} \left( 42-n,n \right) {x}^{41-n}\)
\(\displaystyle \L +45360\,{\it pochhammer} \left( 30-n,n \right) {x}^{29-n} + 8320\,{\it pochhammer} \left( 64-n,n \right) {x}^{63-n}\)
\(\displaystyle \L + 14820\,{\it pochhammer} \left( 26-n,n \right) {x}^{25-n} + 63510
\,{\it pochhammer} \left( 58-n,n \right) {x}^{57-n}\)
\(\displaystyle \L + 36720\,{\it pochhammer} \left( 60-n,n \right) {x}^{59-n} + 18600\,{\it pochhammer} \left( 62-n,n \right) {x}^{61-n}\)
\(\displaystyle \L + 2970\,{\it pochhammer} \left( 66-n,n \right) {x}^{65-n} + 99960\,{\it pochhammer} \left( 34-n,n \right) {x}^{33-n}\)
\(\displaystyle \L + 181480\,{\it pochhammer} \left( 52-n,n \right) {x}^{51-n} + 10\,{
\it pochhammer} \left( 10-n,n \right) {x}^{9-n}\)
\(\displaystyle \L + 256080\,{\it pochhammer} \left( 44-n,n \right) {x}^{43-n} + 67200\,{\it pochhammer} \left( 32-n,n \right) {x}^{31-n}\)
\(\displaystyle \L + 97440\,{\it pochhammer} \left( 56-n,n \right) {x}^{55-n} + 138510\,{\it pochhammer} \left( 54-n,n \right) {x
}^{53-n}\)
\(\displaystyle \L + 136080\,{\it pochhammer} \left( 36-n,n \right) {x}^{35-n} +
246240\,{\it pochhammer} \left( 48-n,n \right) {x}^{47-n}\)

How's that. Understand everything you know about it?.

First time I ever heard of a pochhammer. Also known as a rising factorial.

:wink: :D
 
Wow... good thing we can use "maths" to write an algorithm for a computer to do it for us, eh? :D

Thanks :p
 
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