Expert: Paul Klarreich - 3/6/2010

Question
Hi I need to make 2 general formulas first for D(n)/dx 1/x and the other is for sqrt of (x)
I think I did the first but need help in the second
the nth derivative of 1/x I got is : (-1)^n* n! * x^-(n+1)
for sqrt(x) I got : (-1)^(n+1)*x^(.5-n) but the problem with the coefficient is that it goes in this series (1/2)*(-1/2)*(-3/2)*(-5/2) and it keeps getting -1 for each n I dont know how to express this

Answer
Questioner: hamad
Country: Kuwait
Category: Calculus
Private: No
Subject: General formula of D(n)

Question: Hi I need to make 2 general formulas first for D(n)/dx 1/x and the other is for sqrt of (x)
I think I did the first but need help in the second
the nth derivative of 1/x I got is : (-1)^n* n! * x^-(n+1)

>> Yes, that looks good.

for sqrt(x) I got : (-1)^(n+1)*x^(.5-n) but the problem with the coefficient is that it goes in this series (1/2)*(-1/2)*(-3/2)*(-5/2) and it keeps getting -1 for each n I dont know how to express this
......................
OK, your derivatives go:

D(0) (1/x) =  x^1/2  (original)
D(1) ------= (1/2) x^(-1/2)
D(2) ------= (1/2)(-1/2) x^(-3/2)
D(3) ------= (1/2)(-1/2)(-3/2) x^(-5/2)
D(4) ------= (1/2)(-1/2)(-3/2)(-5/2) x^(-7/2)

Now let's analyze and see if we can relate to n, in:
D(n) ------= (1/2)(-1/2)(-3/2)(-5/2)...(- what/2?) x^(-what/2)

It looks as if the coefficient for D(4) has 3 (-) signs.
Then the coefficient for D(n) should have (n-1) (-) signs.
Write (-1)^(n-1).   [Or, I suppose you could write - (-1)^n, but...]

It looks as if the coefficient for D(4) has 4 factors of  (1/2).
Then the coefficient for D(n) has n factors of  (1/2).

Write (-1)^(n-1)/2^n

It looks as if the coefficient for D(4) has factors  1*3*5, and 5 = 4*2-3
then the coefficient for D(n) should have  1*3*5*...*(4n-3)

How do we express that in factorial form?

          1  3  5  2  4  6   1 2 3 4 5 6
1  3  5 =  ---------------- = ------------
               2  4  6         2  4  6

    6!              (2*3)!
---------------- = --------- =  D(4) coefficient.
2  2  2 ( 1 2 3)    2^3  3!

So try:

 (2n-2)!
--------------
2^(n-1) (n-1)!

as the numerical part.

Put it all together:

(-1)^(n-1) (2n-2)!
-------------------  x^(.5 - n)
2^n 2^(n-1) (n-1)!

and you could do a little simplifying.