Hello I am trying to understand differentiation and I need help with the following problem:
We know that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ verifies, $f(x,x)=x, f(0,y)=0 \forall x,y\in \mathbb R$ and we also know that $f$ is differentiable at the origin. With this information I need to find $$\frac{\partial f(0,0)}{\partial x} $$ Any help would be appreciated.
$$ f'(0, 0)\cdot (1, 1) = \lim\limits_{h\to 0} \frac{f(h, h)-f(0, 0)}{h} = 1 $$ $$ f'(0, 0)\cdot(0, 1) = \lim\limits_{h\to 0} \frac{f(0, h)-f(0, 0)}{h} = 0 $$ $$ \frac{\partial f(0, 0)}{\partial x} = f'(0, 0)\cdot(1, 0) = f'(0, 0)\cdot(1, 1)-f'(0, 0)\cdot(0, 1) = 1 $$ You should be familiar with this: The directional derivative of differentiable function $f$ with respect to the vector $v$ at point $(0, 0)$ is $v\cdot \nabla f(0, 0) $. I wrote it kind of weird, but $f'(0, 0)$ can be identified with $\nabla f(0, 0)$ in this.