I suppose complete understanding of calculus requires 3 different types of understanding
1) intuitive understanding of the concepts
2) mechanical understanding - how to work typical problems
3) precise logical understanding - how to understand and write proofs with statements that involve logical quantifieres like "for each epsilon" and "there exists a delta".
For passing engineering courses that assume you know calculus, you need item 2). It helps to have item 1) also. If you take advanced abstract mathematics, you need item 3).
I'll assume you are asking about item 1).
Think of the derivative as the "rate of change AT at point". The simplest example is velocity. The elementary way to calculate velocity is "Distance divided by time". That only makes sense as an average since there are situations where the velocity is changing a lot during the time interval. (For example, "distance to commute to work divided by time to get to work" is an average of many different rates of travel.) Is it possible to talk about velocity AT A POINT instead of average velocity over a time interval? Yes, that would be what a derivative does. At a point x, you look at smaller and smaller time intervals ( by tradition, they are from x to x + delta x) and you divide the distance moved by the length of these time intervals. Look at the definition of the derivative in your textbooks and notice how it is the limit of a quotient.
Another interpretation of the derivative is the slope of a curve AT A POINT. First, forget what you learned in geometry about the definition of a chord and a tangent. Let's define a "chord of a function" as a line between two points on the function's graph. My calculus teacher illustrated a tangent as follows. He drew the graph of a function on the blackboard. Then labeled two points A and B on the graph. Then he held a ruler so it went between A and B. He held it at point A so it pivoted on one of his fingers. He balanced the ruler at point B by holding the tip of a pencil under it. Then he slowly traced the pencil along the graph so from B to A. The ruler slowly changed its angle since one side of it rested on the pencil. When the pencil reached point A, he declared that the ruler defined the tangent to the curve at point A. (Notice there is nothing in the definition of "tangent to a function" that says it hits the curve at only on point or is perpendicular to some diameter etc. ) Look at the definition of the derivative and try to see how the fraction in the definition defines the slope of this tangent line. (Don't get in the habit of saying that the derivative "is the tangent to the curve", it is "the slope of the tangent to a curve".
If that explains derivative, we can do integration next.
