Studying effective string theory I found the following identity: $$ \ln x = \lim_{s \rightarrow0}\frac{d x^{s}}{ds} . $$ I am however puzzled about its derivation. Naively I would say that $$ \lim_{s \rightarrow0}\frac{d x^{s}}{ds} = \lim_{s \rightarrow0}s x^{s - 1} = 0,$$ which is obviously not the right approach. So my question is, how can I prove the first identity?
Also, I would greatly appreciate if you could help me with giving this questions the right tags.
You are differentiating w.r.t. $x$ but you are supposed to differentiate w.r.t. $s$. The dertivative of $x^{s}$ w.r.t. $s$ is $x^{s} \ln x$.