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? 

You have to be careful with statements like $-\frac{df(x)}{dx}$ being the change in $f(x)$ following a negative change in $x$
Take for example $f(x)=x^2$ at $x=3$ so $f(x)=9$.  Then $\frac{df(x)}{dx}=2x$ which is $6$ when $x=3$.  
A small positive change in $x$ changes $f(x)$ by about the change multiplied by $6$: so for example $f(3+0.01) = 3.01^2= 9.0601$ which is close to $f(3) + 0.01 \times 6$
A small negative change in $x$ also changes $f(x)$ by about the change multiplied by $6$: so for example $f(3-0.01) = 2.99^2= 8.9401$ which is close to $f(3) - 0.01 \times 6$. And it is safer to think of it this way than think of the result being something like $f(3) + 0.01 \times (-6)$.  
In other words, keeping the sign of the change means you do not need to reverse the sign of the derivative   
