What are the different methods to determine the derivative of $f(x)=\sqrt{1+x}$? The methods that I can think of are:
1) Chain rule
2) Binomial series of $f(x)$
3) Through the formula $f\prime(g(x))=\frac{1}{g\prime (x)}$ at a particular point
What are the other methods to determine the derivative of $f(x)$?

Using logarithmic differentiation:
\begin{align}
f'(x) &= f(x) \cdot \frac{d}{dx} \ln f(x)
= \sqrt{1+x} \cdot \frac{d}{dx} \ln \sqrt{1+x}\\
&= \sqrt{1+x} \cdot\frac{1}{2} \frac{d}{dx} \ln(1+x)
= \sqrt{1+x} \cdot \frac{1}{2} \frac{1}{1+x}\\
&= \frac{1}{2\sqrt{1+x}}
\end{align}
