What exactly is a tangent space? I'm self learning differential geometry and got confused by the definition of tangent space. The note I'm using defines tangent space of a smooth manifold $X\subset R^N$ with parametrization $f:U\rightarrow X$ as follow:
$$T_xX=df_0(R^N) $$
where
$$df_x=lim\frac{f(x+th)-f(x)}{t}$$
and$f(0)=x$.
So I suppose that if $f:R^N\rightarrow R$, the tangent space would consist of real numbers. However, when I was reading the example on page 4 of this note (https://folk.ntnu.no/gereonq/TMA4190V2018/TMA4190_Lecture12.pdf), it states that the tangent space of a $f:R^3 \rightarrow R$ is $span\{(-z,0,x),(0,-z,y)\}$ which is not an output of the $df_x$ defined previously. Can someone please explain what exactly a tangent space is? Any help would be appreciated. 

I realized that I mistook the direction of the map in the example mentioned above. 
 A: A tangent space should be thought of as a something straight (a Euclidean space) that is locally (near the point) looks like your space, therefore tangent spaces at a point $x$ of your manifold $X$ and its open subset containing $x$ should be the same (because a tangent space is a local object, i.e. it depends only on local data). 
Then you can take a chart $U$ containing $x$. By definition of a chart, you have a diffeomorphism between $\mathbb{R}^n$ and $U$ (which takes a point $y$ to $x$). Moreover, a tangent space should be something that is invariant under diffeomorphisms and it is functoral, namely if you have a tangent vector at $y$, then you should be able to push it and obtain a tangent vector at $x$. This procedure is called a differential, i.e. a differential is a map on tangent spaces. Then the authors say that to describe a tangent space at $x$ you should take a chart and apply the differential to your map and you obtain a desirable tangent space. Of course, a tangent space at $y$ can be canonically identified with $\mathbb{R}^n$, so all you need is to compute a differential.
