I am having a hard time grasping a concept of derivative intuitively, perhaps due to a lack of a good example of how it can be used in practice. I am looking for an explanation in laymen terms with a practical example that can be deconstructed and would give an idea of how a derivative can be used in practice. I am not looking for mathematical proof or strict mathematical definition.

Here is my current understanding, please point out where it is correct or incorrect intuitively:

Let's say that we have $y$ (dependent variable or output) and $x$ (independent variable or input).
If we have a function of $y=x^{3}$. Does derivative tell us by how much the output of a function (dependent variable $y$) when we change input (independent variable $x$) by a certain amount ($dx$)? In other words, derivative tells us how sensitive the function is to the changes in its input.

P.S. I could not find a satisfactory explanation of this question anywhere on stack exchange.