= Solution
Regard $f$ as a <scalar field> evaluated on a specified <worldline> $x^\lambda(\tau)$. The ordinary multivariable <chain rule> gives
$$
\boxed{\frac{d}{d\tau}f(x(\tau))
=\frac{dx^\lambda}{d\tau}\frac{\partial f}{\partial x^\lambda}
=u^\lambda\partial_\lambda f.}
$$
This is the <directional derivative> along the <four-velocity>. With $x^0=ct$, it reads $df/d\tau=\gamma(\partial_tf+\mathbf v\cdot\nabla f)$. The spatial term is present because the <worldline> moves through the field; it would be lost by identifying a <total derivative> along the particle with a <partial derivative> at a fixed spatial position. For a scalar the contracted expression has the same value in all coordinate systems. For a tensor field the analogous <covariant derivative> also transports its indices.
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