We start with a graph of a continuous function, which means that to draw the graph you don't have to take your pen or pencil off the paper to draw it. Pick any two points on the curve and draw a straight line between them. The slope of the line is calculated from the quotient:
(difference in y coordinates)/(difference in x coordinates).
But on a curve this will not be the slope of the curve (unless it's a straight line), so take two other points on the curve, but this time the points should be closer together. Now the slope created by the two points will be nearer the slope of the curve near the points. As the points move closer together the slope will move closer to the slope of the curve at a single point.
This is the basis of the derivative or differential. The derivative is a limiting value (depending where you are on the curve) and is usually called the gradient at a point. Mathematically it's defined as (f(x+h)-f(x))/h where h→0. You can think of the function f(x) as being the y coordinate at the point x. In precalculus classes, the tutor will lead you to understand the basics behind differentiation by familiarising you with limits and how to calculate them. The derivative above is called df/dx=(f(x+h)-f(x))/h as h→0. If you make a graph of the differential function it may be another curve (or possibly a straight line) so you can repeat the process and obtain the second derivative at a point. The second derivative is usually written d2f/dx2 (or d2y/dx2 if you use y to equal a function of x) for a function f(x). And you can keep differentiating until you get a straight line. Many derivatives never get to a straight line---you can keep differentiating ad infinitum.