# Partial derivative

## 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 Helper
Gold Member

## 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.