Thread: formula for the n:th derivative

Find the formula for the n:th derivative y^(n) for the function y=1/x^(1/2)'

I started by finding out the formula for the functions up to the fourth derivative.
and i came as far as
y = y=1/x^(1/2)
y' = y= -1/2x^(3/2)
y'' =3/4x^(5/2)
y'''=-15/8x^(1/2)
y''''=105/16x^(9/2)

And the pattern which i found is that y^n where n starts from 0. so 0 is the original function.
The nth derivative y^(n) = (-1)^n / ((2^n)*x^(0.5 +n)). The problem here is that i dont find anything that can fix the numerator. I realise the pattern is 1*3*5*7...n. But i dont know how to write a value for this since i cant use ! because its 1*2*3*4 etc. Is it possible to write a factorial equation that skipps whole numbers and only takes odd numbers? And if so, how do i write it mathematically?
Any help is appreciated.

Originally Posted by Zamzen

Find the formula for the n:th derivative y^(n) for the function y=1/x^(1/2)'

I started by finding out the formula for the functions up to the fourth derivative.
and i came as far as
y = y=1/x^(1/2)
y' = y= -1/2x^(3/2)
y'' =3/4x^(5/2)
y'''=-15/8x^(1/2)
y''''=105/16x^(9/2)

And the pattern which i found is that y^n where n starts from 0. so 0 is the original function.
The nth derivative y^(n) = (-1)^n / ((2^n)*x^(0.5 +n)). The problem here is that i dont find anything that can fix the top. I realise the pattern is 1*3*5*7...n. But i dont know how to write a value for this since i cant use ! because its 1*2*3*4 etc. Is it possible to write a factorial equation that skipps whole numbers and only takes odd numbers? And if so, how do i write it mathematically?
Any help is appreciated.

(2n)!! = 2n * (2n-2) *(2n-4) * .... == even numbers
(2n+1)!! = (2n+1)*(2n-1) *(2n-3) * ..... == odd numbers

Originally Posted by yeKciM

(2n)!! = 2n * (2n-2) *(2n-4) * .... == even numbers
(2n+1)!! = (2n+1)*(2n-1) *(2n-3) * ..... == odd numbers

More info

Double Factorial -- from Wolfram MathWorld

If you want a more "standard" formula, for the product of even numbers, you can factor out "2"s:
.

For the product of odd numbers, put in the even numbers: