Is the derivative directional? I was met with a surprising face when I assumed that a derivative is a directional change, i.e. that $$\frac{df(x)}{dx}$$
describes the change in $f(x)$ following an positive change in $x$. Moreover, the negative derivative describes the change in $f(x)$ following a negative change in $x$:
$$-\frac{df(x)}{dx}$$
Am I mistaken? 

Of course since the derivative
$$\frac{df(x)}{dx}$$
represent the rate of change of $f(x)$ for a positive change of $x$ (i.e. for $x$ increasing) then 
$$-\frac{df(x)}{dx}$$
represent the rate of change of $f(x)$ for negative change of $x$ (i.e. for $x$ decreasing).
Observe that the term directional derivative is used when we deal with several variables, in this case for any $\vec v=(a,b)$ we can define the directional derivative as:
$$\frac{\partial f}{\partial \vec v}=\lim_{h\to 0}\frac{f(x_0+ah,y_0+bh)-f(x_0,y_0)}{h}$$
Notably the partial derivatives are the directional derivatives corresponding to the unit vectors $(1,0)$ and $(0,1)$.
Moreover, wheter $f$ is differentiable the following holds:
$$\frac{\partial f}{\partial \vec v}=\nabla f\cdot\vec v=v_1\frac{\partial f}{\partial x}+v_2\frac{\partial f}{\partial y}$$
