= Solution
Interpret the <Stratonovich integral> in its standard extension to continuous <semimartingale> integrands:
$$
\int_0^tY_s\circ dX_s=\int_0^tY_s\,dX_s+\frac12[Y,X]_t.
$$
This extension is needed because $f'(X)$ is generally a <semimartingale> rather than a <local martingale>. Also the introductory prose in the PDF reverses integrand and integrator relative to its displayed definition; the displayed definition fixes the convention used here.
Apply the <Itô formula> to $f'(X)$, using $f\in C^3$:
$$
df'(X_s)=f''(X_s)\,dX_s+\frac12f'''(X_s)\,d\langle X\rangle_s.
$$
The continuous <finite-variation process> in the second term has zero <quadratic covariation> with $X$, while the <quadratic covariation> rule for an <Itô integral> gives
$$
[f'(X),X]_t=\int_0^tf''(X_s)\,d\langle X\rangle_s.
$$
A second application of the <Itô formula>, now to $f(X)$, therefore yields
$$
\boxed{f(X_t)-f(X_0)=\int_0^tf'(X_s)\circ dX_s.}
$$
All unbounded coefficients are handled by stopping $X$ on compact spatial intervals, then removing the <localizing sequence>.
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