Most of the books, notes I have encountered assume that I would know how to prove simple things like this, but it seems like I do not exactly know how. So I tried, whatever I could to show that the map from $\mathbb{R}$ to $\mathbb{R^3}$ defined as $$ f(t) = (\cos(t), \sin(t), t) $$ is an immersion as follows:.
Let $t \in \mathbb{R}$ be any point. The differential of $f$ at $t$ is given as
$$ df_t = \big[ \frac{\partial \cos(t)}{\partial t} \frac{\partial \sin(t)}{\partial t} \frac{\partial t}{\partial t} \big]^t = \big[ -\sin(t) cos(t) 1 \big]^t $$.
I'm not sure how to proceed from here, maybe to conclude that $df$ is indeed an injective map of tangent spaces, I need to show that rank$df = $ rank $\mathbb{R^3}$, which is 3. And the above coloumn matrix also has clearly a rank equal to 3.
Can someone here confirm if the way I've shown a smooth map is immersion is correct or not? I would welcome any hints , corrections to a better approach of this problem.