Let $ \mathcal{C} \subset \mathbb{R}^2 $ be a curve with parametrization $ \overline{r}(t) $. Let $f: \ \mathbb{R}^2 \rightarrow \mathbb{R}$ be a differentiable function given by $f(x,y) $. Consider its restriction on the curve $f\mid _{\mathcal{C}} $ given by $ f(\overline{r}(t)) $.
My question is: is it generally true that $ \frac{\mathrm{d}}{\mathrm{d}t}f(\overline{r}(t))\mid_{t=t_0}=\nabla f(\overline{r}(t_0)) \cdot \frac{\overline{r}'(t_0)}{\Vert \overline{r}'(t_0)\Vert }, $ i.e. the derivative of $f$ on point $ \overline{r}(t_0) $ on the curve is equal to its directional derivative in direction of the tangent line of $\mathcal{C}$ at $ \overline{r}(t_0) $.
$$\frac{d}{dt}f(\bar{r}(t))=\frac{\partial f}{\partial x}\frac{d \bar{r}_1}{dt}+\frac{\partial f}{\partial y}\frac{d\bar{r}_2}{d t}= \nabla f \cdot \bar{r}' \neq \nabla f \cdot \frac{\bar{r}'}{\|\bar{r}'\|}$$