Since $S^1$ is a compact 1-dimensional regular submanifold in $\mathbb{R}^2$ (it's $S^1 = f^{-1}(1)$ for $f : \mathbb{R}^2 \to \mathbb{R}$ given as $f(x,y) = x^2+y^2$), we can find the tangent space for $S^1$ in (1,0) as $$T_{(1,0)}(S^1) = Ker(d(f)_{(1,0)}).$$ Intuitively $T_{(1,0)}(S^1)$ is the plane $\{(1,y): y \in \mathbb{R}\}.$
But we get $$Ker(d(f)_{(1,0)}) = Ker((2x,2y)_{(1,0)}) = Ker((2,0)) = \{(0,y) : y \in \mathbb{R}\}.$$
What it is wrong? Thanks in advance!