<s>[INST] 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 $f(x)$?
 [/INST] From first principles.
\begin{align}
f'(x)&=\lim_{h\to0}\frac{\sqrt{1+x+h}-\sqrt{1+x}}{h}\\
&=\lim_{h\to0}\frac{(\sqrt{1+x+h}-\sqrt{1+x})(\sqrt{1+x+h}+\sqrt{1+x})}{h(\sqrt{1+x+h}+\sqrt{1+x})}\\
&=\lim_{h\to0}\frac{1+x+h-1-x}{h(\sqrt{1+x+h}+\sqrt{1+x})}\\
&=\lim_{h\to0}\frac{1}{\sqrt{1+x+h}+\sqrt{1+x}}\\
&=\frac{1}{2\sqrt{1+x}}
\end{align}

$[f(x)]^2=1+x$. So
\begin{align}
2f(x)f'(x)&=1\\
f'(x)&=\frac{1}{2f(x)}\\
&=\frac{1}{2\sqrt{1+x}}
\end{align}
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