is a notation that means "the derivative of y with respect to x"

when we integrate some function, we say we are taking the anti-derivative of it, because that is what we are doing, we are reversing the process of differentiation. thus when we take the anti-derivative of the derivative of y with respect to x, we go back to y, which was the function we had in the first place.

when we differentiate a function, any constant term in the function goes to zero, since the derivative of a constant is zero. thus we add C. it is the place holder for any arbitrary constant that MAY have been lost when we differentiated y. that's why the function y we obtain after integrating must have a C attached to it.Why do we need a constant integration on the right but not left hand side???

let's see.What would happen if y=5 when x=0??

when y = 5 and x = 0 we have:

5 = 5(0)^2 + C

=> C = 5

thus we can find our arbitrary constant given this information. so the original function y was actually, y = 5x^2 + 5

this is a huge question. it would be nice to use some graphs as illustrations, but i can't bother. i will say this though. the graph of the derivative of a function is a graph that describes the slope of the function at any point, and as far as polynomials go, the graph of a derivative always has one less turn than the original graph. how does the graph of the derivative describe the slope. well, there are several ways in which it can do that. where the slope of a function is positive, the graph of the derivative will be above the x-axis, where it is negative, the graph of the derivative will be below the x-axis, where it is zero, the graph of the derivative will cut the x-axis. and there are many other things to be extracted based on how the graph of the erivative is curved and what-notand what is the graphical meaning of this differential equation and its solution??