What is a derivative?

To put it simply, derivatives are slopes. 

A derivative can also be written as f '(x). 

If x here is a particular point, such as f ' (3), then the result will be a slope of the function when it is at x =3

If it is only f ' (x), then this represents the derivative of the whole function. Ie, f ' (x) is a function that can be used to find any slope on the original function, and from that, the exact point on the original function can be found. 

You can also call f ' (x) the instantaneous slope of points. 

If we pick two points on the function f (x), point x and point x+h, is it possible to find the slope between these two points.

m = [ f (x+h) - f (x) ] / ( x + h - x )

    = [ f (x+h) - f (x) ] / h

Because the slope m is the derivative f ' (x), we can write....

This is also another definition for a "derivative".

[to be continued]

A definition of Limits

Ever saw one of those graphs that either reach into infinity, and then appear again, or those graphs that randomly skip orders? I'm talking about these...

Just a quick reference note, black dots mean "There is a point here", and white dots mean "There is no point here but it comes so close to the point that there is almost no difference - but there still actually is no point here".

This is where the concepts of limits come in. So, what are limits?

Let's visualize this so we can understand this definition better.
Here, we have a function that is a linear equation, except for when it is at (x, L), where there is no point available. This is represented by a white dot. At this point, we can say that the Limit is L.

Hence, we can say that



If we pic any points ϵ units away from L on either side, we will have an x that is δ units away from point "x". We know that we do not have L if we input "x" into the function f(x), because there is no "point" there. L is the limit, or the closest approximation we can get to as you input an x into f(x) that is infinitely close to "x".

If δ is the length between the picked point and "x", no matter how small δ is, these two points will differ, but they are infinitely close so that for any difference doesn't matter, and the limit will be considered "L".