Prove that $$ 2x < \log{\frac {1+x}{1-x}} < 2x \left[{\frac {1+x^2}{3 (1-x^2)}}\right] $$ where $0 < x <1.$ Hence, deduce that $$ e < \left({1+ \frac 1n}\right)^{n+\frac12} < e^{1+\frac {1}{2n (n+1)}} $$
I have solved the first part by taking derivative but I'm not able to deduce the second part from the result obtained.