Landau defines
$\log x = \lim\limits_{k \to 0} {x^k-1 \over k}$
I wanted to prove the elemental properties of the logaritm with this, namely:
- $\log xy = \log x +\log y $
- $\log x^a = a\log x $
- $1-\dfrac 1 x\leq\log x \leq x-1 $
- $\lim\limits_{x\to 0}\dfrac{\log(1+x)}{x}=1 $
- $\dfrac{d}{dx}\log x = \dfrac 1 x$
I proved them all, however, in the last case I did this
$\eqalign{ & \frac{d}{dx}\log x = \lim \limits_{h \to 0} \frac{\log \left( x + h \right) - \log x}{h} \cr & = \lim\limits_{h \to 0} \lim \limits_{k \to 0} \frac{\left( x + h \right)^k - x^k}{kh} \cr & = \lim \limits_{k \to 0} \frac{1}{k}\lim \limits_{h \to 0} \frac{\left( x + h \right)^k - x^k}{h} \cr & = \lim \limits_{k \to 0} \frac{1}{k}\lim \limits_{h \to 0} kx^{k - 1} = \lim \limits_{k \to 0} \lim \limits_{h \to 0} x^{k - 1} = x^{ - 1} } $
Since I'm not familiar with multivariable calculus, I don't know how to justify this. What could work here?