I'm getting lost trying to figure out what seems to be a simple question. I've looked in Shafarevich and Görtz & Wedhorn but the points are usually dealt with using schemes, which I haven't gotten to yet.
Here, I will only consider affine varieties over an algebraically closed field $k$. I know that tangent spaces are functorial in the sense that morphisms of varities $\phi:V \rightarrow W$ imply maps $T_P\phi:T_PV \rightarrow T_{\phi(P)}W$ and I'd like to know when exactly this allows to describe all of the tangent space.
Here's the particular case I'm on: consider the map $f:\mathbb{A}_k^1\rightarrow \mathbb{A}_k^n$, $x \mapsto (x^{a_i})$ where the $a_i$ are integers with gcd 1. We can show that the map is finite and hence that the image $C$ is an irreducible affine curve, the defining ideal $I_C$ of which is the kernel of the corresponding ring map $k[X_1,\dots,X_n] \rightarrow k[X]$, $P \mapsto P((X^{a_i}))$. We can define the tangent space by noticing some things on generating elements of the kernel.
My question: is there a faster way to derive the tangent space directly from the parametrisation we have, just like in Differential geometry ?