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In the limit definition of the derivative $f'(x)=\lim_{u\to x}\frac{f(u)-f(x)}{u-x}$, I understand that $(x, f(x))$ is the point where you want to find the instantaneous slope. However, I see a lot of visual proofs of the derivative where they draw the point $(x, f(x))$ on a curve called $f$ with coordinate axes that are already called $x$ and $f(x)$. I don't get how $x$ can stand for a coordinate of a point as well as an axis at the same time. The same thing goes with $f(x)$.

Also, how can I graph the point $(u, f(u))$ on the curve $f$ with axes that are called $x$ and $f(x)$. Yes, I could change the axes to be called $u$ and $f(u)$ but then how could I graph the point $(x, f(x))$ on a coordinate plane with axes $u$ and $f(u)$? Points $(x, f(x))$ and $(u, f(u))$ have to be graphed on the same coordinate plane but I don't see how you can do that.

I understand the concept of a derivative as a limit but the notation and symbols and letters used to write out the general derivative formula throws me off. Sorry if it seems like I am asking ridiculous questions.

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  • $\begingroup$ In your notation, since you are taking the limit as $u\rightarrow x$, the fixed point in which you are computing the slope of the tangent line is the pair $(x,f(x))$. By the way, maybe it is a clearer notation the following one: $lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = f'(x)$ $\endgroup$ – Dadeslam Jun 9 at 7:30
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    $\begingroup$ But how can you graph the point $(x+h, f(x+h))$ on a plane with the axes $x$ and $f(x)$. $\endgroup$ – user565207 Jun 9 at 7:48
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    $\begingroup$ On a plane with axes $x$ and $f(x)$, you can only graph the point $(x, f(x))$ $\endgroup$ – user565207 Jun 9 at 7:51
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    $\begingroup$ It is just a matter of notation, we are calling the axis and even the point with the $x$ letter. If you prefer just denote the x-axis with the letter $x$ and the limit to define the derivative in terms of $c$ as follows : $lim_{h\to 0} \frac{f(c+h)-f(c)}{h}=f'(c)$, where here we are computing the derivative at the point $(c,f(c))$ which is fixed. It is just an arbitrary point on the curve which is the image of the function, with input $c$ in the domain and output $f(c)$ in the arrival space. $\endgroup$ – Dadeslam Jun 9 at 8:02
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    $\begingroup$ Yes exactly, maybe call $x$ and $y$ the axes if it clarifies the context. $\endgroup$ – Dadeslam Jun 9 at 9:52
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I hope you know the informal definition of a function, and understand the associated notation:

Informal Definition of Function:

Given a set $A$ and a set $B$, a function $f$ from $A$ into $B$ is a "rule" which assigns to each element $\xi$ in $A$, a unique element of the set $B$. This unique element is denoted by the special symbol $f(\xi)$. In this case, we call $A$ the domain and $B$ the target space for $f$.

The above sentences above are summarised by the following symbols: " $f:A \to B$ "

Now, let's specialise to the case where you are interested in, namely, when $A=B = \mathbb{R}$ (the set of real numbers). Then, if you pick a particular number $\xi \in \mathbb{R}$, we say $f$ is differentiable at $\xi$ if the limit \begin{equation} \lim_{u \to \xi} \dfrac{f(u) - f(\xi)}{u - \xi} \end{equation} exists. If it exists, we give this limit the special name $f'(\xi)$.

Now, as far as definitions go, that's all there is to differentiability, and I assume you've seen all of this. But what I would like you to focus on is my choice of words. I've used phrases like "special symbol"/"the special name" etc to try to emphasize the distinction between a concept/idea and a name for that concept/idea. The former is the important thing, while the latter is just a manner of speaking, and it is important to not confuse the two (this may sound like a silly statement, but it seems to be the source of your confusion). In this discussion, the key concept is "differentiability at a particular point $\xi$", while the notation can be anything like $f'(\xi)$ or, $\dot{f}(\xi)$ or $\dfrac{df}{dx}(\xi)$ or $\dfrac{df}{dx} \bigg|_{x = \xi}$.

(As an aside: the last two notations are very nice to manipulate, but they are very misleading notations, and it can straight up be confusing, especially if you're computing the derivative at $x$; they are often misused by people who understand what they are talking about, but too lazy to explain/ poorly explain what it means. If what I said confused you, then take that as a reason to use either the first or second notation.)

Ok now we can try to interpret the concept of differentiability from a geometric perspective. To each function $f: \mathbb{R} \to \mathbb{R}$, we can try to pictorially understand it by drawing its graph. I think the terminology "$x$-axis", "$y$-axis" and "$f(x)$-axis" is very bad for beginners because firstly, what's so special about $x$? why not call it $y$ and $f(y)$ axis, or $\xi$ and $f(\xi)$ axis? Secondly, suppose we have two functions $f$ and $g$. Then if you draw the graphs of both functions together, it clearly doesn't make sense to speak of "$f(x)$-axis"... because why not call it the "$g(x)$-axis"? Therefore, my suggestion to you is to simply think of it as the "horizontal axis" and "vertical axis", and call them "the domain of functions", and "target space of functions". For example, if you only have one function $f: \mathbb{R} \to \mathbb{R}$ in a discussion, call the horizontal and vertical axes "domain of $f$" and "target space of $f$" respectively. If you have two or more functions, say $f,g,h: \mathbb{R} \to \mathbb{R}$, then simply call the horizontal and vertical axes "domain of $f,g,h$" and "target space of $f,g,h$" respectively.

(In the above paragraph, it is only due to several hundreds of years of convention that we often think of the horizontal axis as being the domain of the function, and think of the vertical axis as being the target space of the function, rather than the other way around; so don't think too hard about this.)

Now, if you choose a particular point $\xi$ of the domain, the function $f$ gives us a unique point $f(\xi)$ in the target space; thus we get a unique "tuple" of points $(\xi, f(\xi))$, which we can draw as a point in the plane, as shown below (notice the order in which things happened; you first pick a point of the domain, then the function gives you a unique point of the target, and only after that can you "draw" this point in the plane):

If you pick a different point $\eta$ in the domain, you will again get a unique point $f(\eta)$ in the target space, so you can draw the point $(\eta, f(\eta))$.

If yet again you pick a different point $\alpha$ in the domain, you will again get a unique point $f(\alpha)$ in the target space, so you can draw the point $(\alpha, f(\alpha))$, and so on: If you do this for every point of the domain, then you finally get the graph of the function $f$:

In the above paragraphs, notice how I intentionally avoided the terms "$x$-axis" and "$f(x)$-axis", so now there can be no confusion of

I don't get how $x$ can stand for a coordinate of a point as well as an axis at the same time. The same thing goes with $f(x)$.

Also, how can I graph the point $(u,f(u))$ on the curve $f$ with axes that are called $x$ and $f(x)$. Yes, I could change the axes to be called $u$ and $f(u)$ but then how could I graph the point $(x,f(x))$ on a coordinate plane with axes $u$ and $f(u)$?

The way to think about the axes is as "the domain of the function" and "the target space of the function", so it DOES NOT matter what you label points on the axes as. If you want, you can label a point on the horizontal axis (the domain) as $@$, and the corresponding point on the vertical axis (the target space) as $f(@)$, or $!$ and $f(!)$, or $\ddot{\smile}$ and $f(\ddot{\smile})$, or even $\ddot{\frown}$ and $f(\ddot{\frown})$.

With this more precise terminology, I hope it allows you to clarify what people mean when they say things like "$f'(x)$ can be seen as the limit of the slope of the secant line joining the points $(u,f(u))$ and $(x,f(x))$, as $u\to x$". Once again, I want to emphasize that you should distinguish between a concept/idea vs names/terminology we use to express that idea, because if you don't you'll often miss the forest for the trees.

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