# Derivative ### How does a function affect nearby points? Let us have a function, $f$, that is applied to $x$. We then look at $f$ evaluated at $x=a$. The question of certain interest is: > How does $f$ impact the points **nearby** $x = a$ (such as $x = a + \epsilon$ and $x=a-\epsilon$)? A great visualization can be found [here](https://youtu.be/wCZ1VEmVjVo?t=409). For instance, consider that we are interested in $f(x) = x^2$. We want to see how, when $f$ is applied to $x = 3$, it impacts nearby points. We can do the following; let us have a $x = 3$ and the nearby point $x=3.1$. We can apply $f$ to each: $f(3) = 9 , \;\; f(3.1) = 9.61$ So, we started with our points being $0.1$ apart ($3.1 - 3 = 0.1$), and they are now $9.61 - 9 = 0.61$ apart. If we take that ratio we see that: $\frac{\text{How far are nearby points after applying }f?}{\text{How far are nearby points before applying }f?} = \frac{9.61 - 9}{3.1 - 3} = 6.1$ Now if we *zoom in* on our point $x=3$ and look *even closer neighbors*, we will see that the above ratio converges to $6$. Which of course is simply the *derivative* of $x^2$ at $x=3$: $\frac{d}{dx}(x^2) = 2x \longrightarrow 2(3) = 6$ So, the **derivative** can be thought of as: >* *How a function impacts nearby points in space*. >* The *scaling factor* of $f$ near a point $x=a$ Even more generally we can view the derivative as: > The derivative describes how a function **transforms space** in the neighborhood around a certain point. > > Note: The derivative is a function itself (so it is a function describing a function). This view of course depends on us having a good understanding of [Space, Transformations and Descriptions](Space,%20Transformations%20and%20Descriptions.md) and [Space Transformations](Space%20Transformations.md). ### How sensitive is a function? An easier way to think about the derivative is simply: > It measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). ### How does integration factor in? It is helpful when thinking of integration to think about *mass*. Consider we have a rod and we wish to find its mass. Unfortunately there are different densities at different points along the rod. Let us describe the position along the rod with $x$, and the density at a given point with $f(x)$. And of course recall that $mass = density \times volume$, where here volume is simply a $1$-dimensional length. To find the mass of the rod, we can chop it into smaller pieces and then add them all up. If we chop the rod into pieces of small enough length, then we can approximate it to have uniform density along a single piece:  For each rod we simply take the length (the same for each rod) and multiply it by the constant/uniform density for that specific rod in order to get the mass. We then add up all of these masses and get a closer and closer approximation to the true mass. This process is denoted by the integral: $\int_a^b f(x) dx$  With that said, I still feel like at this point there is some missing intuition around the integral and its relationship with the derivative. A key idea is as follows: > * A function $f$ can be described (and hence reconstructed) via its *local* behavior. > * We can describe $f$ in this way via another function, the *derivative*. We can call this $f’$: > $\overbrace{f \rightarrow f'}^{\text{derivative}}$ > * Via the derivative (local information) we can perform a full reconstruction to the original function. In this way we take *local* information and compose it to achieve a *global* description: > $\overbrace{f' \rightarrow f}^{\text{integral}}$ > * This reconstruction is done via the *integral*. > * If we know where we start (the initial conditions), and we meticulously keep track of the local behavior (derivative information; how is space transformed in a local area) we can reconstruct our original function via integration. Why do I think this needs to be clearly articulated? Well, the problem is that if we start with a function $f$ and then calculate its derivative, $f’$, it is very clear that integration can take us from $f’$ to $f$. However, what does the integral of $f$ mean? This is why I wanted to articulate the above bullets. The integral of $f$, say it is $g$, is a function. But, if we want to know how the derivative and the integral relate, we need to focus on $f$ and $f’$. The derivative takes us from $f$ to $f’$, and the integral takes us from $f’$ back to $f$. But if our main goal is to analyze the derivate and integral in terms of $f$ (i.e. what happens when we apply the derivative to $f$, and what happens when we apply the integral to $f$), the integral requires we introduce a new function $g$. We cannot simply stick with just $f$ and $f’$. Whereas if we only want to analyze the derivative and integral, and we aren’t too picky about what functions we look at, we are able to focus only on two functions, $f$ and $f’$, and we don’t need to think about $g$. ### Change of variables   Key idea: the focus is on the *mass* - *not the density*. And that is indeed the key idea you are trying to grok here! TODO: ensure this is corrrect --- Date: 20220721 Links to: [Mathematics MOC](Mathematics%20MOC.md) Tags: #review References: * [What is Jacobian? | The right way of thinking derivatives and integrals - YouTube](https://www.youtube.com/watch?v=wCZ1VEmVjVo&t=493s)