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Material Derivative of the Gradient of a Scalar Field

Let $f$ be a scalar field that is continuous and does not vary along the flow, that is $D_t(f)=0$ where $D_t=\partial_t+\vec u\cdot\nabla$ where $\vec u$ is the incompressible velocity field (i.e $\text{div}(\vec u)=0$. I am to show that for this $f$, $D_t(\vec \omega\cdot\nabla f)=0$ where $\vec\omega=\text{curl}(\vec u)$.

I am able to simplify this using Einstein Summation notation to be $D_t(\vec\omega\cdot\nabla f)=(\vec\omega\cdot\nabla \vec u)\cdot\nabla f$ by using the fact that $D_t\,\vec \omega=\vec\omega\cdot\nabla\vec u-\vec\omega\,\text{div}(\vec u)$.

What I get hung up on is the step where you must find $D_t(\nabla f)$. It seems to me that this should be identical to $\nabla D_tf$. Obviously this is incorrect, but I don't understand why...

Thank you for any replies