/sci/ - Science & Math

I know the standard algebraic definition of a linear transformation / linear operator. But every single source I checked simply states that "the derivative is a linear operator since it satisfies the definition of the linear operator". Really? But "linear" means it is a line. Where is the line? Whats the geometric interpretation? Why is it never ever mentioned in any books? For example ln'(x) = 1/x. Why is this a linear operation? Is it because a derivative at every point is the tangent line ? So the derivative is an operator that maps ever point to the slope of the tangent line at the point? That's the "line" in "linear"? A local linear approximation?