Confused about variables This is probably a very dumb question but after trying to review some calculus after years not using it, I am confused by variables in the equation for a tangent line.
So I watched the very first lecture on calculus by MIT ( https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006) and at around $0:31$, I am very confused about the $X$ and $X_0$ he uses.
Basically, there is a function $y = \frac{1}{x}$ and we are trying to find the area under all triangles formed by the tangent line of this function. Differentiation was used in order to find the function $y=\frac{1}{x^2}$ which shows the slope of the original function for any $X$. I perfectly understand the logic of these steps etc. but I am extremely buffled by separating the $X$ and $X_0$ -> the lecturer even pointed out that it often confuses people but didn´t elaborate more on it.
Basically, the problem for me starts right when trying to find the equation for the tangent line, given the derivative we found. The equation for a line is $Y - y_0 = m(X-x_0)$
For some reason, I am tempted to just write: $Y - y_0 = (\frac{1}{x^2})*(X-X)$ which obviously is a nonsense since we´d obtain a zero in the parenthasis. But I just cannot justify in my mind the fact that we use x0 instead of $X$ to model the slope $m$. I´d say that the slope will be a function of $X$ and therefore, I´d never ever think of using something like the $x_0$.
Can please someone help to clarify this for me? thank you!
 A: Well, the underlying concept is simple. We take a certain point $(x_0, y_0)$ lying on the given curve. At this point, slope of tangent is obviously given by:
$$m=\left(\frac {dy}{dx}\right)_{x=x_0}$$
Hence, for your given curve, at $x=x_0$,  $$m=-\frac {1}{{x_0}^2}$$
Thus, keeping in mind that $(x_0, y_0)$ are not variables, rather parameters, we can write equation of tangent:
$$\frac {y-y_0}{x-x_0}=m=-\frac {1}{{x_0}^2}$$
Here $x,y$ represent the variables in which we represemt the line.The slope, $m$, and one point on the line, that is, the point of tangency $(x_0,y_0)$, parameters for us.
A: I'm getting nostalgia for lecturing with chalk and erasers like those...
You write, “we are trying to find the area under all triangles formed by the tangent line of this function.”  But there's not just a single tangent line to the curve, is there?  There are infinitely many—any point on the curve is going to produce a tangent line to the curve at that point.  The professor says:

We're going to calculate the areas of triangles, and you could ask yourself, “In terms of what?”

He proceeds to mark a generic point on the curve with coordinates $(x_0,y_0)$.  Now the goal is much more refined: find the area of of the triangle enclosed by the axes and line tangent to the curve $y=1/x$ at $(x_0,y_0)$, as an expression involving (“in terms of”) $x_0$ and $y_0$.
Why not label that point $(x,y)$ instead of $(x_0,y_0)$? Because the professor wants to use $x$ and $y$ as variables for the equation of the curve, and the equation for the tangent line(s).  As you discovered, if you try to do both, your algebraic expressions collapse to nothing as a result of this overloading.
After the example is finished the professor points out what he sees as a big obstacle to learning calculus: understanding what the variables are, and when.  As Ritam Dasgupta notes in his answer, $x_0$ isn't changing in the course of this problem—it's unknown, but it's fixed.  In this particular problem the area of the triangle is a constant, no matter what tangent line is used, but you can imagine curves where this area depends on $x_0$.  What if you wanted to the largest triangle? Well, now you do treat $x_0$ like a variable and differentiate the area function in terms of it.
Don't assume that if this example is confusing, you're not meant to learn calculus.  I'm not sure this example is the best choice for the first day of Calculus I. MIT led the way on open courseware back in 2006, but by now there are many, many calculus lectures online.  Use that to your advantage and find one that's to your liking.
