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? 

Just to give some completion here, if the function $y=f(x)$ passes through the point $(x_0,y_0)$, and has derivative $f'(x_0)$ at that point, the equation of the tangent at $(x_0,y_0)$ is:$$y-y_0=f'(x_0)(x-x_0)$$ This is the line which passes through $(x_0,y_0)$ and has gradient $f'(x_0)$.
This relates to the fact that $$f'(x_0)=\lim _{x\to x_0}\frac {f(x)-f(x_0)}{(x-x_0)}=\lim _{x\to x_0}\frac {y-y_0}{(x-x_0)}$$
