About intertemporal elasticity of substitution
To close this one: As I wrote in my initial comment, one should consider the derivative with respect to $s$, not $c(s)$. This first means that anything containing $t$ has a zero derivative. So we have  

for the numerator

$$\frac {d\ln(c(s)/c(t))}{ds} = \frac {c'(s)}{c(s)}$$

and for the denominator

$$\frac{d[\ln(U'(c(s))/U'(c(t)))]}{ds} = \frac {U''(c(s))\cdot c'(s)}{U'(c(s))}$$

Forming the quotient of the two we get after simplifying and adding the minus sign the desired

$$\lim_{s \to t}\sigma = -\frac{U'(c(t))}{U''(c(t))c(t)}$$