Partial and total derivative of a multivariable function Given a function $f(x,y,t)$, is it correct to say
$$\frac{d f}{d x} = \frac{\partial f}{\partial x} \text{ ?}$$

The function $f$ is of three variables, so you should be using $$\frac{\partial f}{\partial x},$$ or $f_x$ or $f_1$ to denote the partial derivative of $f$ with respect to $x$.
Here $df/dx$ has no meaning because $f$ is a function of three variables. You would only use this notation (or $f'$) if $f$ was a function of $x$ alone. (You might also use $f'$ to denote the Jacobian matrix  of partial derivatives for a function of many variables.)
