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Tue 11 Apr, 2006 03:35 pm
I'm currently working on a calculus assignment where we find a pattern and create a y^n = formula I am stuck with the top:
y=1/x-1
y'=-1/(x-1)^2
y''=2/(x-1)^3
y'''=-6/(x-1)^4
y''''=24/(x-1)^5
You can obviously see that the bottum is (x-1)^n+1 but the top i am struggling with finding a correct n value which would account for all cases. Any help would be appreciated!
The numerator is n! and it changes sign each time, so you need a (-1)^n in there. The denominator is just a power of (x-1), but it is one higher than the level of deriviative you are working on. Put those together and you have your answer.
it's impossible for the numerator to be n(-1)^n as the fourth deriviative is 24 and 4(-1)^4 is 4...
24 is 4! 1*2*3*4=24 the next one will have a 5! constant.
for the odd even (-1)^n is odd for n odd, even for n even
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