-емый/-имый

1. Formation
These participles are usually formed by adding specific suffixes to the present tense stem of imperfective verbs (perfective verbs cannot form present participles because they lack a present tense). They are primarily formed from transitive verbs, but there is an important exception for intransitive verbs that govern the instrumental case (e.g., руководить → руководимый, управлять → управляемый).

• 1st Conjugation: Add the suffix -ем- (or sometimes -ом-) plus adjective endings. A helpful trick is to take the 1st-person plural (мы) form and add -ый (Note: see Section 4 for exceptions like -авать/-знавать verbs, which use the infinitive stem). For the -ем- suffix, the stress usually matches the present tense form.
  читать → мы читаем → читаемый (readable/being read); рисовать → мы рисуем → рисуемый.
• 2nd Conjugation: Add the suffix -им- plus adjective endings. Use the 1st-person plural (мы) form as the base to avoid consonant mutations (present passive participles do not undergo consonant mutation). Warning: While the 'мы' form gives the correct letters, it will give you the wrong stress for verbs with mobile stress (e.g., мы любим but любимый). For stress: it falls on the stem if the verb has fixed stem stress, and on the -им- suffix for mobile or end stress.
  видеть → мы видим → видимый (visible); любить → мы любим → любимый (beloved/favorite).

2. Participle vs. Adjective
In formal writing, if an agent is mentioned in the instrumental case, it acts as a true participle (a process). Without an agent, it often becomes a 'potential' adjective (a quality).
Example: проверяемые учителем работы (papers being checked by the teacher) vs. незабываемый опыт (an unforgettable experience).

3. Usage and Register
This form is common in academic, legal, and literary Russian. In daily speech, Russians usually prefer a relative clause with который or a reflexive passive. However, many specific words have become standard everyday adjectives, such as любимый (favorite) or необходимый (necessary).

4. Irregularities and Gaps

• Verbs ending in -авать, -знавать, and -ставать (like давать, узнавать, доставать) form this participle from the infinitive stem rather than the present tense stem: дава-ть → даваемый, узнава-ть → узнаваемый, достава-ть → доставаемый.
• The suffix -ом- (which always takes the stress) is used for 1st conjugation verbs ending in -ти or -чь, plus the verb искать: вести → ведомый, влечь → влекомый, искать → искомый (which is standard in modern technical and mathematical contexts).
• The word движимый (moved by / driven by) is an important exception: it is spelled with -им- even though the modern verb двигать is 1st conjugation.
• Many common verbs (пить, бить, писать, ждать) cannot form this participle.

5. Short Forms
Like adjectives, present passive participles have short forms (e.g., читаем, читаема, читаемо, читаемы) that function as predicates.