The correct answer is: D. No improvement
The sentence “The detectives followed on several clues but failed to find the murderer” is grammatically correct and conveys the intended meaning clearly. The word “on” is used in the sense of “in the direction of” or “towards”. In this case, the detectives are following the clues in the direction of the murderer. The word “on” is also used in the sense of “in pursuit of” or “searching for”. In this case, the detectives are following the clues in search of the murderer.
The other options are not as clear or concise. The word “up” could be interpreted as meaning “to a higher level” or “to a more advanced stage”. This would not make sense in the context of this sentence. The word “through” could be interpreted as meaning “from one end to the other” or “from beginning to end”. This would also not make sense in the context of this sentence. The word “by” could be interpreted as meaning “using” or “by means of”. This would not make sense in the context of this sentence.
Therefore, the correct answer is: D. No improvement