Let's look at the use of monotonic transformations of utility functions (which I guess is the most frequent occurrence of this concept in econ).
Let $u: \mathbb{R}^n_+ \to \mathbb{R}$ be a utility function. We say that $g: \mathbb{R}^n_+ \to \mathbb{R}$ is a monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$:
$$
u(x) \ge u(y) \iff g(x) \ge g(y).
$$
Let $D \subseteq \mathbb{R}$. Then $h: D \to \mathbb{R}$ is called strictly increasing if for all $x, y \in D$:
$$
x \ge y \iff h(x) \ge h(y).
$$
There's the following result:
Th Let $D$ be the range of $u$, i.e. $D = u(\mathbb{R}^n_+)$. Then $g$ is a monotonic transformation of $u$ iff there exists a strictly increasing function $h: D \to \mathbb{R}$ such that $g(x) = h(u(x))$.
proof: ($\leftarrow$) If $g(x) = h(u(x))$ and $u(x) \ge (>) u(y)$ then $h(u(x)) \ge (>) h(u(y))$ so $g$ is indeed a monotone transformation of $u$.
($\rightarrow$) For the reverse, define
$$
h(z) = g(x) \text{ whenever } u(x) = z.
$$
First we need to check that $h$ is indeed a function. As such, let $x, y$ be such that $u(x) = u(y) = z$. We need to show that $g(x) = g(y)$. Indeed as $g$ is a monotone transformation, we have that $g(x) \ge g(y)$ and $g(y) \ge g(x)$ so $g(x) = g(y)$. To see that $h$ is strictly increasing, let $z \ge (>) w$ and let $x$ and $y$ be such that $u(x) = z \ge (>) w = u(y)$. Then as $g$ is a monotone transformation $h(z) = g(x) \ge (>) g(y) = h(w)$. $\blacksquare$
To see the connection with the derivatives we have the following:
Th if $g: D \to \mathbb{R}$ is differentiable, $D$ is convex and if $g'(x) > 0$ for all $x$ in $D$, then $g$ is strictly monotone.
proof: Let $x \ge (>) y$ then by the mean value theorem there is a $c \in [x,y]$ such that:
$$
g(y) - g(x) = g'(c)(y - x) \ge (>) 0.
$$
Remark: as @Michael Greinecker said, the reverse is not true, there are strictly increasing functions whose derivative is zero at some points, like $g(x) = x^3$.
But how is that true for a negative monotonic transformation such as g(x)=−x where g′(x)=−1?
In principle we could define the reverse notions. Say that $\tilde g$ is a "negative" monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$:
$$
u(x) \ge u(y) \iff \tilde g(x) \le \tilde g(y).
$$
Notice that $\tilde g$ reverses the ranking given by $u$.
Next, we can look at functions $\tilde h: D \to \mathbb{R}$ that are strictly decreasing:
$$
x \ge y \iff \tilde h(x) \le \tilde h(y).
$$
We have the following:
Th The function $\tilde g$ is a negative monotone transformation of $u$ iff there exists a strictly decreasing function $\tilde h: D \to \mathbb{R}$ such that $\tilde g(x) = \tilde h(u(x))$.
The proof is similar to the proof for the monotone/strictly increasing case.
Remark: In economics, "negative" monotonic transformations are not really used. We would like the functions $u$ and $g$ to give the same ranking over all bundles $x \in \mathbb{R}^n_+$ then they should be monotone transformations of each other. If you use a negative monotonic transformation, you are reversing the order.
