Try doing what Hurkyl suggested: look at a few cases first.
for n= 1 (uv)'= u'v+ uv': the product rule
for n= 2 (uv)"= ((uv)')'= (u'v+ uv')'
= (u'v)'+ (uv')'= (u"v+ u'v')+ (u'v'+ uv")
= u"v+ 2u'v'+ uv"
Hmmm that is NOT u"v+ u'v'+ uv" as your formula claims!
for n= 3 (uv)"'= ((uv)")'= (u"v+ 2u'v'+ uv")'
= (u"v)'+ 2(u'v')'+ (uv")'
= (u"'v+ u"v')+ 2(u"v'+ u'v")+ (u'v"+ uv"')
= u"'v+ 3u"v'+ 3u'v"+ uv"'
Keep going until you recognize a pattern, then use induction to prove it.
