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Each component is easily transformed but the sum is not:

FourierTransform[f''[x], x, k]
FourierTransform[f'[x], x, k]
-k^2 FourierTransform[f[x], x, k]
-I k FourierTransform[f[x], x, k]
FourierTransform[f''[x] + f'[x], x, k]
FourierTransform[f'[x] + f''[x], x, k] (*!!*)

LaplaceTransform behaves correctly:

LaplaceTransform[f''[x] + f'[x], x, k]
-f[0]-k f[0]+k LaplaceTransform[f[x],x,k] + k^2 LaplaceTransform[f[x],x,k] - f'[0]

Just in case. Win7 V9.0.1.0

As acl has noticed on chat, Distribute would be quick fix but not sure how stable for more complicated cases:

Distribute[ FourierTransform[f''[x] + f'[x], x, k]]
-I k FourierTransform[f[x],x,k]-k^2 FourierTransform[f[x],x,k]
share|improve this question
Related: FourierTransform and Partial Derivatives? – Jens Jan 14 '14 at 18:38
Just wrote a "shell" for FourierTransform here, which (I think) is a little more stable than the Distribute approach. – xzczd Jan 9 '15 at 12:34

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