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

If $y = x^{n}$ , 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 $x^{n}$ . From my first year analysis I think I recall $x^{n}$ is defined to be $e^{n \ln x}$ for $x$ real. You now have to show that

$\lim_{δ x \to 0} \frac{e^{n \ln (x + δ x)} - e^{n \ln x}}{δ x} = n x^{n - 1}$

I guess you then expand then now expanding

$e^{[(n δ x) / (x)]} = 1 + \frac{n δ x}{x}$

for small $δ x$ .

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

Hope this helps

Geoff M.