If I'm asked to differentiate a function that is simply of the form: $ f(tx_1, tx_2, ... tx_n) $ (i.e., the function is not "defined" to be particularly anything), how should I go about this? I want to differentiate with respect to $ t$.
Context: Let's say we have a homogeneous function of with degree $ k $.
Thus,
$$ f( tx_1, tx_2, ... tx_n) = t^k f(x_1, x_2, ... x_n) $$
Differentiating on both sides with respect to $ t$, we get,
$$ x_1 f_1 (tx_1, tx_2, ... tx_n) + ... + x_n f_n (tx_1, tx_2, ... tx_n) = k \cdot t^{k-1} \cdot f(x_1, .. x_n) $$
On the RHS, why we differnetiate with respect to to get $ k \cdot t^{k-1} $. Because the function on the RHS is a constant (i.e., it doesn't depend on t, we write it as it is)
On the LHS, why do we have: $ x_n f_n(tx_1, tx_2,\dots, tx_n)$ for each term?
On the RHS, you just apply the usual rules of single variable calculus. The factor $t^k$ becomes $k t^{k-1}$ and $f(x_1,\dots,x_n)$ is constant with respect to $t$, so you can just keep it.
For the LHS you are looking for the chain rule for multivariable functions. If $f(x_1,\dots,x_n)$ is a function of $n$ variables and you have $n$ functions $z_1(t),\dots,z_n(t)$ each of one variable $t$, then
$$ \frac{\mathrm d}{\mathrm dt} f(z_1(t),\dots,z_n(t)) = \frac{\partial f}{\partial x_1}(z_1(t),\dots,z_n(t)) \cdot \frac{\mathrm dz_1}{\mathrm dt}(t) + \dots +\frac{\partial f}{\partial x_n}(z_1(t),\dots,z_n(t)) \cdot \frac{\mathrm dz_n}{\mathrm dt}(t) $$
In your example you have $z_i(t) = tx_i$ so that $\frac{\mathrm dz_i}{\mathrm dt}(t) = x_i$.