Algebraist's definition of the tangent space of a manifold

Question

By the "algebraist's definition" of the tangent space of manifolds, can we say that the partial derivative $d/dx$ belongs to the the tangent space of $S^1$? It feels strange, but I can't see why it shouldn't be true.

Answer 1

If $f:S^1\rightarrow \mathbb R$ is a smooth function, then how can you differentiate w.r.t. the variable $x$? This is impossible! $$ {f(x+h,y)-f(x,y) \over h} $$ makes no sense, since $(x+h,y) \notin S^1$