Clearifiny basic calculus?

Hey guy, i was introduced to calculus a few days ago and wanted to ask if im right. What ive learned:

The gradient of a chord is the rate of change of a graph. As h tends to 0, that point is the gradient of a tangent line which gives the gradient of a single point on a graph. Btw, is h a proper notation that is used to represent x+h - x (x2-x1) or is it a random one off thing? Anywho so the process of this is called differentiation and the product is called the gradient function. So if im not mistaken the gradient function gives the gradient at any one point of a graph.

Plz point out any mistakes and 1 last question, if the gradient function of f(x) is f'(x) then is the gradient function of dy/dx is d^2y/dx^2

Sorry i would have asked my teacher but hes a total tool. Ignores everyone with a question. I swear they pick teachers off the street. TYVM