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If we write something like:

$\partial_a X_{\mu} \partial^a X^{\mu}$

Does that mean the first derivative is only applied to the first X?

($\partial_a X_{\mu})( \partial^a X^{\mu}$)

Or is the first derivative applied to the object $X_{\mu} \partial^a X^{\mu}$, such that second derivatives would appear?

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It is definitely an ambiguous notation, but one that is quite conventional. You should interpret it as: $(\partial_a X_\mu)(\partial^a X^\mu)$. For instance, often the Klein-Gordon Lagrangian is written as: $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi + \cdots $$ which should be interpreted as: $$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)( \partial^\mu \phi)+ \cdots $$

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In my experience, the correct interpretation is $$ \partial_\mu X \partial^\mu X = (\partial_\mu X)( \partial^\mu X). $$

Total derivatives are usually written clearly as $$ \partial_\mu(\ldots). $$

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The conservative answer is: It depends on context. Different authors mean different things. See also this Phys.SE post.

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