pin690
Member

What is the nth derivative of nx^n?

thecalculuskid
Member

The first derivative is
(n)(n)x^(n1)=(n^2)x^(n1).
The second derivative is
(n^2)(n1)x^(n2).
The third derivative is
(n^2)(n1)(n2)x^(n3).
Interestingly, the nth derivative would be
(n^2)(n1)(n2)(n3)….[n(n2)][n(n1)]x^(nn)
where x^(nn) 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)….(n3)(n2)(n1)n^2
n^2 can be broken down into (n)(n).
Hence the nth derivative is
n! times n or ….. nn!

practicaldave
Member

Good answer 🙂 @thecalculuskid

thecalculuskid
Member

So, what do you think pin?
Do I get the point? 😉
