Partial Derivative: Difference & Chain Rule

[Niles]

Homework Statement

Hi

Say I have a function $f(x(t), t)$. I am not 100% sure of the difference between
$$\frac{df}{dt}$$
and
$$\frac{\partial f}{\partial t}$$
Is it correct that the relation between these two is (from the chain rule)
$$\frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\frac{dx}{dt}$$

 

[LCKurtz]

Homework Statement

Hi

Say I have a function $f(x(t), t)$. I am not 100% sure of the difference between
$$\frac{df}{dt}$$
and
$$\frac{\partial f}{\partial t}$$
Is it correct that the relation between these two is (from the chain rule)
$$\frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\frac{dx}{dt}$$

It is easy to be confused by the ambiguity of $\frac{\partial f}{\partial t}$ symbol. If you write the expression instead as $f(u,v)$ where $u = x(t),~v=t$ you would write$$
\frac{df}{dt} = f_u\frac {du}{dt} + f_v\frac{dv}{dt}=f_u\frac{dx}{dt}+f_v\cdot 1$$You wouldn't normally talk about $\frac{\partial f}{\partial t}$ as though $f$ depended on another variable also. But as the chain rule gives, you need the partials of $f$ with respect to each of its arguments. If you understand that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial t}$ in this setting mean the partials of $f$ with respect to its first and second arguments, you should be OK.