Agree with Jim - ‘doucement’ adds a factor of ‘reducing/avoiding disturbance’ or ‘being careful’ - and is a word young French children can hear quite regularly !

Lentement is more specific to a comment on ‘speed’.

However, as doing something slowly is often required to do something gently, there is some overlap in contextual meaning, in French and in English translation.

Hi Maeve,

The two adverbs have a certain synonymy but it is a matter of context as to which is the most appropriate.

If the meaning is gently / softly / delicately then  "doucement" otherwise for slowly "lentement" would probably be more apt.

Hope this helps.

Jim