Tangent space to a level curve

Question

Let $\Omega$ be an open set of $\mathbb{R}^n$.
Let $f:\Omega\to \mathbb{R}$ a differentiable function, and $X=\{ x\in \Omega \, | \, f(x)=c\}$ a level curve. If $v$ is a tangent vector to $X$ in $a$, there exists $\gamma : ]-\varepsilon , \varepsilon [ \to A$ a function of class $C^1$ such that $\gamma(0)=a$ and $\gamma'(0)=v$. The function $f \circ \gamma$ is constant, hence $(f\circ\gamma)'=0$, which is equivalent to : \begin{equation} \mathrm{d}f_{\gamma(t)}(\gamma'(t))=0 \end{equation} For $t=0$, we have $\mathrm{d}f_a(v)=0$, so every tangent vector of $X$ in $a$ is in the kernel of $\mathrm{d}f_a$.
I would like to know if we have the converse : is every vector in the kernel of $\mathrm{d}f_a$ tangent to $X$ in $a$?