Posted by Lisa on Sunday, March 10, 2013 at 5:37pm.
let f(x)= x/x1
find f'(x) f ''(x) and a formula for f ^ (n) * x.
I found the first and second derivatives but do not know how to make a general equation for this. I have not learnt the Taylor or Maclaurin Series either. Thank you.

Calculus Derivative Taylor Series?  Reiny, Sunday, March 10, 2013 at 5:51pm
your first derivative should have been:
y' = 1(x1)^2
then y'' = 2(x1)^3
y''' = 6(x1)^4 > the third derivativ
y'''' = 24(x1)^5 > the 4th derivative
y''''' = 120(x1)^6 > the 5th derivative
did you notice that the numbers 1 , 2, 6, 24 , 120 are factorials ?
Did you notice that the alternate ±
did you notice that the exponent is (n+1) for the nth derivative, e.g. , for the 4th derivative the exponent is 5
how about a general derivative expression from the above?

Calculus Derivative Taylor Series?  Lisa, Sunday, March 10, 2013 at 6:06pm
Yes I did get those as my derivatives. Thank you very much for your help Reiny.
I did not know which patterns to look for.
I think that it would be (1)^n for the alternating neg. and pos.?
Then n! for the factorial? And lastly x^ (n+1) for the power of x in the denominator?
I got: f^ (n) (x)= (1)^n * n! * x^ (n+1) Is this right?
Thank you very much again. I never knew what patterns to help for. Now I know how to do such problems :)

Calculus Derivative Taylor Series?  Lisa, Sunday, March 10, 2013 at 6:06pm
*look

Calculus Derivative Taylor Series?  Reiny, Sunday, March 10, 2013 at 6:27pm
good job
especially the (1)^n part, good of you to notice that if n is even , the result has to be positive, and if n is odd we need a negative.
(1)^n will do that

Calculus Derivative Taylor Series?  Anonymous, Sunday, March 10, 2013 at 6:53pm
Thank you!
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