Jerky gear change on diesel automatic Toyota Yaris?

I have bought a second-hand 2008 diesel automatic Toyota Yaris. The gear change between first gear and second gear is very jerky. There is a lag and then the gear changes, it the same way a new driver might make a mess of this changing manually.

I've had autos before and even a Yaris auto before and it was smoother than this. (Very good actually.) I've never had a diesel before. Is this to be expected with a diesel, maybe because of the way it revs? Or are the automatic boxes on the new Yaris not as good as the old ones? Could it be a fault (cheap I hope) to fix? (Software adjustment?)