Topic: nth derivative

nth derivative

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February 23, 2012 at 5:14 pm #432
pin690 Subscriber	What is the nth derivative of nx^n?
February 23, 2012 at 8:06 pm #438
thecalculuskid Subscriber	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
practicaldave Subscriber	Good answer @thecalculuskid
February 23, 2012 at 8:14 pm #443
thecalculuskid Subscriber	Thanks practicaldave.
February 23, 2012 at 8:15 pm #444
thecalculuskid Subscriber	So, what do you think pin? Do I get the point?
February 23, 2012 at 8:17 pm #445
2plus2 Subscriber	Yes, great answer.