By Woon Khang Tang (P3742) on Wednesday, April 25, 2001 - 08:10 pm:

If y = xn, then dy/dx will be n x(n-1). I know that by using first principles, there's a pattern, but how to prove that it's true for all real values of n?

By Geoff Milward (Gcm24) on Wednesday, April 25, 2001 - 10:16 pm:

For n real we have to define what we mean by xn. From my first year analysis I think I recall xn is defined to be en ln x for x real. You now have to show that
limdx® 0 en ln(x+dx)-en ln x
dx
 =n xn-1
I guess you then expand then now expanding
e[(ndx)/(x)]=1+ ndx
x
for small dx.

It is considerably easier for n integer as one can just expand out (x+dx)n = xn +n xn-1 dx+.., but I guess this is what you already knew. I think the real difficulty here is the formal definition of xn for n real.

Hope this helps

Geoff M.