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- chain rule order of differentiation in the product

This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example

## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ##

and not

## 2 \bf{v} \cdot \nabla \bf{v} ##

Is there an obvious way to see / think of this from the chain rule, say in 1-D, preferably through looking at the limit definition?

Thanks

## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ##

and not

## 2 \bf{v} \cdot \nabla \bf{v} ##

Is there an obvious way to see / think of this from the chain rule, say in 1-D, preferably through looking at the limit definition?

Thanks