= Solution
The displayed rational expression has domain $\mathbb R\setminus\{0\}$; its value at zero is undefined despite the printed $\mathbb R\to\mathbb R$ notation. On that domain, the <power rule> gives
$$
\boxed{\begin{aligned}
f'(x)&=2x-4+\frac4{x^2}-\frac2{x^3},\\
f''(x)&=2-\frac8{x^3}+\frac6{x^4},\\
f^{(3)}(x)&=\frac{24}{x^4}-\frac{24}{x^5},\\
f^{(4)}(x)&=-\frac{96}{x^5}+\frac{120}{x^6},\\
f^{(5)}(x)&=\frac{480}{x^6}-\frac{720}{x^7}.
\end{aligned}}
$$
These <derivatives> are smooth on every interval avoiding zero, including all intervals used below.
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