What exactly is the derivative of a function? I am very new to calculus. I have only just started it in school and when I have looked at other people explaining this question they use a lot of terminology I am unfamiliar with and it gets very overwhelming. I just want to know, is the derivative of a function of a graph a way of finding the graph’s gradient at (x, y)? Also, what exactly is the difference between $\frac{\mathrm dx}{\mathrm dy}$ and f(x) = … and when should I use the different functions. Thank you for any help I appreciate it so much.
 A: Too long for a comment: The derivative is many things. It starts with

The derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.

(which I guess looks familiar to you) and continues all the way through

The derivative of a real-valued function $f$ in a domain $D$ is the Lagrangian section of the cotangent bundle $T^*(D)$ that gives the connection form for the unique flat connection on the trivial $\Bbb R$-bundle $D ×\Bbb R$ for which the graph of $f$ is parallel.

(which is probably not familiar to you at all, and honestly, a bit overwhelming to me). There are dozens upon dozens of different actively used notions of derivative, all trying to capture the same concept from different angles. What you think the derivative is depends entirely on your current level, what math you are currently doing, and your current mood.
In my opinion, it is for this reason nearly impossible to truly answer the question "What exactly is the derivative of a function?" The best you will get is a list of different definitions like the two above. None of them wrong, but not all of them suitable to you, and none of them the "true" notion of derivative.
A: $f'$ is known as the derivative or gradient function, and $f'(x)$ tells us the gradient of $f$ at the point $x$. For instance, if $f(x)=x^2$, then $f'(x)=2x$, meaning at any point $(x,f(x))$ that lies on the graph of $f$, the gradient is $2x$.
The graph of $f$ can also be described as the set of points in the $x$-$y$ plane that satisfy $y=x^2$. Then, we can write $\frac{dy}{dx}=2x$  (not $\frac{dx}{dy}=2x$), which has the same meaning as $f'(x)=2x$, but tries to capture a slightly different idea. While the notation $f'(x)=2x$ emphasises that $f'$ is the gradient function, the notation $\frac{dy}{dx}=2x$ tries to capture the geometric content of the derivative. Roughly speaking, $\frac{dy}{dx}=2x$ means that if we start at the point $(x,y)$, and move a small distance $\Delta x$ along the graph, then the change in $y$-coordinate $\Delta y$ will be about $2x$ times the change in $x$-coordinate $\Delta x$. More precisely, $\frac{dy}{dx}$ is the limit of $\frac{\Delta y}{\Delta x}$ as $\Delta x$ approaches $0$:
$$
\frac{dy}{dx}=\lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x \to 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} \, .
$$

$f$ is a function—not a graph. A function can be thought of as a machine that takes each input to a particular output. We commonly denote an arbitrary input to the function using the letter $x$ (but there's no particular reason to—it's just convention). So $f(x)=x^2$ means that the function $f$ takes each number to the square of that number. For instance, $f(1)=1$, $f(47)=47^2$, and $f(\pi)=\pi^2$. Technically, $f$ is the function, whereas $f(x)$ is the value of the function $f$ at $x$. In practice, it is common to say that $f(x)$ is the function.
The letter $x$ is just being used to refer to any particular point on the $x$-axis.
I don't think I completely understand your next question, but $\frac{dy}{dx}=2x$ and $f'(x)=2x$ are both very common notations, and you can use whichever you like (the first is called Leibnizian notation, and the second is Lagrange notation or prime notation). Generally, if a function is defined by an equation $y=\ldots$ then it is customary to write $\frac{dy}{dx}=\ldots$ If a function is defined by $f(x)=\ldots$, then the derivative is denoted by $f'(x)=\ldots$ Some people express a strong preference for one of these notations over the other. Personally, I think both of them are useful in their own way.
$f'$ is the gradient function, and $f'(x)$ is the gradient of $f$ at $x$. For instance, if $f(x)=x^2$, then $f'(x)=2x$. So $f'(5)=10$ means "if you draw a tangent to the graph of $f$ at the point $(5,25)$, then the gradient of that tangent will be $10$". If you plot the graph of $f'$ in the $x$-$y$ plane, then at each point $(x,y)$ of $f$, the value $y$ tells us the gradient of $f$ at $x$.
You can find the tangent to the graph of $f$ at the point $(x_1,y_1)$ by working out $f'(x_1)$. For example, if $f(x)=3x^2$, then $f'(x)=6x$. So the tangent to $f$ at the point $(10,300)$ must have gradient $f'(10)=60$. If the line $\ell$, given by $y=mx+c$, is tangent to $f$ at $(10,300)$, then it must be the case that $m=60$. Also, $\ell$ passes through $(10,300)$, and so $300=60(10)+c$, meaning that $c=-300$. Therefore, $\ell$ has equation $y=60x-300$. In general, the tangent to the graph of $f$ at the point $(x_1,y_1)$ has the equation
$$
y = f'(x_1)(x-x_1)+y_1 \, .
$$
