1. ## Directional derivative

Let $\displaystyle E$ be a subset of $\displaystyle \mathbb{R}^n, f:E\rightarrow \mathbb{R}^m$ be a function, $\displaystyle x_0$ be an interior point of $\displaystyle E$, and let $\displaystyle v$ be a vector in $\displaystyle \mathbb{R}^n$. I want to show if $\displaystyle f$ is differentiable at $\displaystyle x_0$, then$\displaystyle f$ is also differentiable in the direction $\displaystyle v$ at $\displaystyle x_0$, and

$\displaystyle D_vf(x_0)=f'(x_0)v,\ \ \ \ \ \ \$

where $\displaystyle D_vf(x_0):=\lim_{t\rightarrow 0, t\geq 0}\frac{f(x_0)+tv)+f(x_0)}{t}$

Thanks

2. What is your definition of
1) Differentiable

2) Differentiable in the direrction v?

3. Originally Posted by HallsofIvy
1) Differentiable

2) Differentiable in the direrction v?
I say $\displaystyle f$ is differentiable at $\displaystyle x_0$ with derivative $\displaystyle L$ if we have

$\displaystyle \lim_{x\rightarrow x_0,x\in E-{x_0}}\frac{||f(x)-(f(x_0)+L(x-x_0))||}{||x-x_0||}=0$
where $\displaystyle ||x||$ is the lenght of $\displaystyle x$ as measured in the $\displaystyle l^2$ metric

I say $\displaystyle f$ is differentiable in the direction $\displaystyle v$ at $\displaystyle x_0$ if the limit

$\displaystyle \lim_{t\rightarrow 0;t>o,x_0+tv\in E}\frac{f(x_0+tv)-f(x_0)}{t}$ exists.

4. Originally Posted by bram kierkels
I say $\displaystyle f$ is differentiable at $\displaystyle x_0$ with derivative $\displaystyle L$ if we have

$\displaystyle \lim_{x\rightarrow x_0,x\in E-{x_0}}\frac{||f(x)-(f(x_0)+L(x-x_0))||}{||x-x_0||}=0$
where $\displaystyle ||x||$ is the length of $\displaystyle x$ as measured in the $\displaystyle l^2$ metric

I say $\displaystyle f$ is differentiable in the direction $\displaystyle v$ at $\displaystyle x_0$ if the limit

$\displaystyle \lim_{t\rightarrow 0;t>o,x_0+tv\in E}\frac{f(x_0+tv)-f(x_0)}{t}$ exists.
So what happens if you put $\displaystyle x = x_0+tv$ in the first of those definitions?