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1. Origins, structure, and basic meaning

• Structure: 揮 has 12 strokes and the on-reading き. 発揮 (はっき) uses small っ; 指揮 (しき) reads き after し.
• Basic usage: 発揮 concerns demonstrating abilities or results; 指揮 concerns directing a performance or group activity.

2. Common pitfalls and similar expressions

• Demonstrating ability and directing activity
  • 練習の成果を発揮した means demonstrating the results of practice. 表現力を発揮してください asks someone to use their expressive ability.
  • 演奏を指揮した means directing a performance. 隊長の指揮のもとで means under the captain's command, so 指揮 is not limited to music.