nth derivative

Forums Calculus nth derivative

This topic contains 5 replies, has 4 voices, and was last updated by Avatar of 2plus2 2plus2 2 years, 9 months ago.

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February 23, 2012 at 5:14 pm #432
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pin690
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What is the nth derivative of nx^n?

February 23, 2012 at 8:06 pm #438
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thecalculuskid
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The first derivative is

(n)(n)x^(n-1)=(n^2)x^(n-1).

The second derivative is

(n^2)(n-1)x^(n-2).

The third derivative is

(n^2)(n-1)(n-2)x^(n-3).

Interestingly, the nth derivative would be

(n^2)(n-1)(n-2)(n-3)….[n-(n-2)][n-(n-1)]x^(n-n)

where x^(n-n) is x^0, which is equal to 1, so that term kind of “diappears”….

The other factors can be looked upon in the reverse order and you would get:

(1)(2)(3)….(n-3)(n-2)(n-1)n^2

n^2 can be broken down into (n)(n).

Hence the nth derivative is

n! times n or ….. nn!

February 23, 2012 at 8:12 pm #442
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practicaldave
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Good answer :) @thecalculuskid

February 23, 2012 at 8:14 pm #443
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thecalculuskid
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Thanks practicaldave.

February 23, 2012 at 8:15 pm #444
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thecalculuskid
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So, what do you think pin?

Do I get the point? ;)

February 23, 2012 at 8:17 pm #445
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2plus2
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Yes, great answer.

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