Is the derivative directional?

Question

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?

Answer 1

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:

$$\frac{\partial f}{\partial \vec v}=\nabla f\cdot\vec v=v_1\frac{\partial f}{\partial x}+v_2\frac{\partial f}{\partial y}$$

Answer 2

Yes, it is directional. The same way that when we say that a function is "increasing", we are seeing it from left to right.

Answer 3

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