If we have a multivariate function \( f(x,y) \) where \( x=x(t) \) and \( y=y(t) \) are functions of a variable \( t \), we have that the gradient of \( f \) with respect to \( t \) (without the explicit unit vector components)
$$ \frac{df}{dt} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial t} \end{bmatrix}=\frac{\partial f}{\partial x} \frac{\partial x}{\partial t} +\frac{\partial f}{\partial y} \frac{\partial y}{\partial t}. $$