Derivative of a function is the equation of the tangent line? So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the same equation throughout the curve? 


So what exactly is a derivative?

The derivative is instantaneous (i.e. at any given precise moment in time) the rate of change of a dependent variable (usually $y$) with respect to the independent variable (usually $x$).
For straight lines, the derivative is simply the slope or gradient of the line.
For curves, whose gradient (slope) is constantly changing, the derivative gives the instantaneous rate of change at a given point; i.e. the slope of the tangent at that point.
There is one and only one tangent (that's what tangent means in Latin- "touching"to a curve at any given point.
The tangent is a straight line that just touches the curve.

So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve?

In a sense, yes. There's the equation of the curve, $y=f(x)$, which could be any function ($x^2, e^x, \ln(x), \tan(x),$ etc.), and then there's the equation of the tangent line at a given point.
If you choose a different point on the curve, not only will you change the gradient of the tangent at the point, but also the equation of the tangent.

How can the equation of the tangent line be the same equation throughout the curve?

This is not true.
I'll show you with a sketch of an arbitrary function $y=f(x).$

I've just picked 3 random points on the curve (denoted by purple dots).
At each of these points, I've drawn the tangents to the curve at that specific point. As you can see, these are clearly different lines, so they can't possibly have the same equation.
