Let $f: {\mathbb R}^n \to \mathbb R$ be a ${\mathbb C}^1$ function. The graph of $f$ is the surface $M :=\{(x, f(x)) \in {\mathbb R}^n \times \mathbb R | x \in {\mathbb R}^n\}$. Given an arbitrary point $p :=(x_0, f(x_0)) \in M$ on the graph of $f$, how to express the tangent space $T_pM$ in terms of the gradient $\nabla f(x_0)$? Here $\nabla f(x_0) \in {\mathbb R}^n$.
My approach is to use the theorem: "At a regular point $x_0$ of the surface $S$ defined by $h(x)=0$, the tangent plane is equal to $M = \{y : \nabla h(x_0)y = 0$". And then consider the surface $M$ in the above question as zero set $g(x, z)=f(x) -z = 0$. Here $g(x, z)$ can be the constraint which is same as $h$ in the theorem.
Is my approach correct to get the formula of $T_pM$ in terms of the gradient $\nabla f(x_0)$? Is there any better, more direct way?