Why do we take a derivative? I don't have a fundamental  understanding of what taking a derivative means and I don't understand why would we do it. I just know how to mechanically do it.
If $f(x)$ is a (real) function (of a real variable), then its derivative $f&#39;(a)$ at a point $a$ measures the sensitivity of $f$ to small changes in $x$ around $a$. For example, suppose $x$ is "the height of the sun in the sky" and $f(x)$ is "the length of the shadow cast by this tree." Then $f&#39;(a)$ is a measure of how sensitive the length of the shadow is to a small change in the height of the sun if the height is around $a$. Experimentally, $f&#39;(a)$ will be small if the sun is high in the sky ($a$ is large) while $f&#39;(a)$ will get larger as the sun gets lower in the sky ($a$ is small). The derivative gives us a way to quantify this observation.

I decided not to use the term "rate of change" because I think some students have heard this term so many times that they have become immune to it, so perhaps the above will make more sense.
if you given a function and you asked to find the behave of the given function then you may take the derivative to the given function, but if you given a behave of any function you need to take the integral of the behave function, to find the function of the given behave.