Partial and total derivative of a multivariable function Given a function $f(x,y,t)$, is it correct to say
$$\frac{d f}{d x} = \frac{\partial f}{\partial x} \text{ ?}$$
 A: 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.)
A: For a ternary function $f(x,y,t)$, the expression $\dfrac{df}{dz}$ would usually be read as the total derivative of $f$ with respect to the exogenous argument $z$. In general, you have for this total derivative that $$
  \frac{df}{dz} =
  \frac{\partial f}{\partial x} \cdot \frac{dx}{dz} +
  \frac{\partial f}{\partial y} \cdot \frac{dy}{dz} +
  \frac{\partial f}{\partial t} \cdot \frac{dt}{dz}.
$$ 
For $z=t$, if neither $x$ nor $y$ depends on $t$, then $\frac{dx}{dt} = \frac{dy}{dt} = 0$, and since $\frac{dt}{dt} = 1$ you indeed get that $$
\frac{df}{dt} = \frac{\partial f}{\partial t}.
$$
