The difference between $\frac {df}{dt}$ and $\frac {\partial f}{\partial t}$ What is the difference between $\frac {df}{dt}$ and $\frac {\partial f}{\partial t}$, I understand this if I have the coordinates, but not with time.
 A: If $f$ is a function of one variable, $\dfrac{df}{dt}$ is the derivative of that function.  If $f$ is a function of several variables, $\dfrac{\partial f}{\partial t}$ is the partial derivative of that function with respect to the variable $t$.
Now sometimes (especially in physical applications) you have a quantity that can be can be thought of in different ways, sometimes as a function of one variable, sometimes as a function of several.  
For example, the energy of an object may depend on position as well as time, so from that point of view it's a function of four variables $E = E(x,y,z,t)$, and you might write $\dfrac{\partial E}{\partial t}$ for the partial derivative with respect to $t$ (with $x, y, z$ held constant).
On the other hand, if the object is moving in a certain way, its position coordinates $x, y, z$ will themselves be functions of $t$.  We then might write $E = E(x(t), y(t), z(t), t)$, a function of only one variable $t$, and the derivative of this would be $\dfrac{dE}{dt}$.
A: You are asking about the difference between the total derivative
$$
 \frac{df}{dt},
$$
and the partial derivative
$$
 \frac{\partial f}{\partial t}.
$$

If the function depends upon a single variable,
$$
 f \colon \mathbb{R} \mapsto \mathbb{R}
$$ 
then the total derivative is used. The total derivative of $f$ at the point $p$ can be computed using
$$
  \frac{df}{dt}\Big\lvert_{x=p} = \lim_{t\to p} \frac{f(t+p) - f(t)}{t-p}
$$

When the function depends upon more than a single variable, say $m$ variables,
$$
 f \colon \mathbb{R}^{m} \mapsto \mathbb{R}
$$ 
then the partial derivative is used. 
Consider
$$
 f(x,y,t)
$$
The partial derivative of $f$ at the point $p$ in the direction of $t$ can be computed using
$$
  \frac{\partial f(x,y,t)}{\partial t}\Big\lvert_{t=p} = \lim_{t\to p} \frac{f(x,y,t+p) - f(x,y,t)}{t-p}
$$
while all the other variables, $x$ and $y$ are fixed.

When the function depends upon variables which are functions of $t$, like
$$
 f(x(t),y(t),t),
$$
the the total derivative is computed using the chain rule:
$$
  \frac{df}{dt} = 
\frac{\partial f}{\partial t} + 
\frac{\partial f}{\partial x} \frac{\partial x}{\partial t} +
\frac{\partial f}{\partial y} \frac{\partial y}{\partial t}
$$
A: $\frac{df}{dt}$ is used when $f$ is solely based on $t$. For instance, you would use this on $f(t)=2t^2$.
$\frac{\partial f}{\partial t}$ is used when $f$ depends on multiple variables, but you only care about how $f$ changes when $t$ changes. In other words, assume every variable except $t$ is a constant. For instance, if $f(t,x,y)=t^4xy+x^y-y^3t$, then $\frac{\partial f}{\partial t}=4t^3xy-y^3$.
