Tangents and Calculus

Say we have a function f(x), and we want to define analytically the line tangent to the graph at the point (x,f(x)). Then we say define the tangent line as the line passing through that point with slope equal to the derivative at that point, namely 
             f(w)-f(x)
f'(x) = lim  ---------.
        w->x    w-x
If we drew secant lines drawn from the point (x,f(x)) to other points (w,f(w)) which approach it along the curve, this derivative would be precisely the limit of the slopes of those lines.