The formal definition of directional derivatives

Question

So, let's assume that I have a line $y=mx$ in cartesian systems and that I want to find the directional derivative of the function $f(x,y)$ along this line. The position vector $\vec{r}$ and the unit vector in this direction would be $$\vec{v}=\frac{1}{\sqrt{1+m^2}}\pmatrix{1\\m}\quad \quad \vec{r}=\pmatrix{x\\y}$$ Formally the directional derivative in this direction would be defined as $$\partial_vf=\lim_{\lambda\to 0}\frac{f(\vec{r}+\lambda\vec{v})-f(\vec{r})}{\lambda}=\lim_{\lambda\to0}\frac{f(x+\frac{\lambda}{\sqrt{1+m^2}},y+\frac{\lambda m}{\sqrt{1+m^2}})-f(x,y)}{\lambda}$$ Now, some authors define directional derivatives as $\partial_vf=\nabla f\cdot\vec{v}$, hence in accordance to them $$\partial_v f=\lim_{\lambda\to0}\left[\frac{f(x+\lambda,y)-f(x,y)}{\lambda}.\frac{1}{\sqrt{1+m^2}}-\frac{f(x,y+\lambda)-f(x,y)}{\lambda}.\frac{m}{\sqrt{1+m^2}}\right]$$ Where am I going wrong? Kindly suggest a direction...

Answer 1

Recall the Taylor expansion of a differentiable function in $\mathbb{R}^n$

$$ f(\mathbf{x}+\mathbf{a})=f(\mathbf{x})+\mathbf{a}\cdot\mathbb{\nabla} f(\mathbf{x})+\mathcal{O}(|\mathbf{a}|^2)$$

use $\mathbf{a}=\lambda\mathbf{v}$ to get

$$\frac{f(\mathbf{x-\lambda v})-f(\mathbf{x})}{\lambda}=\mathbf{v}\cdot\mathbb{\nabla} f(\mathbf{x}) + \mathcal{O}(\lambda^2) $$

take the limit $\lambda\rightarrow 0$ to get the result