1. The problem statement, all variables and given/known data

Suppose an element of a string, called [tex]\[\triangle x\][/tex] with T being the tension.
The net force acting on the element in the vertical direction is

I know what small-approximation is, but I suspect there is a definitive reason to why we choose sin ~= tan and not sin ~= delta. But y/x is arctan.. if we are talking about that.. So what is it?

To be more clear, the reason we use partial is because the function contains two variables, x and t, right?

Well, for small-angle approximations you can set sin(θ) ≈ tan(θ) or sin(θ) ≈ θ, depending on which one is more useful for the particular calculation you're doing. In this case it appears that they want to use the derivative dy/dx, which is equal to the tangent of the angle, so it's more useful to choose tangent.

Right, and because you want to take the derivative of y with respect to only x, leaving t constant.

LOL I am so stupid. tan = sin/cos, and I always thought x/y. It was opposite / adjacent, which makes dy/dx.

After reading a bit on derivative on Wiki,

According to the book

1. So why do the physicists imagine this "infinitesimal displacement"?

2. So in essence, the rate of change, dy/dx gives the rate of change. If we interpret it in dy/dx form, we have the slope of a tangent line. I see the relationship between dy/dx and tan, but how do I see the relationship between the slope of the tangent line and tan?

1. As opposed to a finite displacement or something? I'm not sure I understand what you're confused about here. Generally speaking, that's just one way to think about a derivative, you move an infinitesimal amount in the x direction and see how much your function changes in the y direction. (If it were a finite displacement, you might have different slopes at different points within the interval.)

2. Well why do you think they call it the tangent function? Try this: just draw a straight line on a graph, and draw a triangle to figure out the slope as [itex]\Delta x/\Delta y[/itex]. Then using the same triangle, find the angle between the line and the x-axis.