Learn Russian: Он руководит обоими отделами.

The verb руководить (to manage, lead, or direct) always requires its object to be in the instrumental case. In Russian, you don't manage something (accusative); you manage with or by means of something (instrumental).

How do we get the form обоими for "both"?

Since отдел (department) is a masculine noun, we start with the masculine form for "both", which is оба. To make it agree with the instrumental plural noun отделами, оба changes to обоими.

What would this look like if the noun was feminine, like "companies" (компаниями)?

For feminine nouns, the base word for "both" is обе. Its instrumental form is обеими. So, "He manages both companies" would be Он руководит обеими компаниями. Notice the subtle spelling difference: обоими for masculine/neuter versus обеими for feminine.

Is there a trick to remembering the meaning of руководить?

Yes! It is made of two very common roots: рука (hand) and водить (to lead or guide). So, to manage or direct something in Russian is literally to "lead it by the hand."