The tangent line, nature's best approximator to a graph

[Ostronomos]

I haven't quite ceased with my mathematical musings. Like the medieval theologians before me, I believe mathematics can unveil the mind of God.

On a graph of a function y=f(x), the average rate of change of y with respect to x between x=x1 and x=x2 is Δy/Δx=y2-y1/x2-x1.

It's interesting that we can choose any two arbitrary points on the secant line and it will always give the same answer for slope. Calculus gives us new tools that help you predict the behavior of functions.

Now that I've firmly established my intellectual limits, I hope to weigh in on a more thorough treatment of these concepts in the near future.

[confused2]

y=f(x), the average rate of change of y with respect to x between x=x1 and x=x2 is Δy/Δx=y2-y1/x2-x1.

So we draw a line from x1,y1 to x2,y2 .. as you say the slope is Δy/Δx=y2-y1/x2-x1.
To get a grip on this let's let the function be y=x^2

y1 is (obviously) just x1^2
Let x2 be x1+Δx so y2 is (x1+Δx)^2
multiplying it out..
(x1+Δx)^2 = x1^2 + 2x1Δx + (Δx)^2
So (from your Δy/Δx=y2-y1/x2-x1)
Δy/Δx=[(x1^2 + 2x1Δx + (Δx)^2) -x1^2]/[(x1+Δx)-x1]
Subtracting all the things we can subtract we get
Δy/Δx=[2x1Δx + (Δx)^2]/[Δx]
Now we do something crazy..
we say as Δx get smaller .. and smaller .. and smaller .. the small thing multiplied by another small thing .. the (Δx)^2 .. gets even smaller until foop! she's gone and we say we don't even need to think about .. so we get
..as Δx tends to zero..
Δy/Δx=[2x1Δx + foop]/[Δx]
or
Δy/Δx=[2x1Δx]/[Δx]
cancelling the Δx
Δy/Δx=2x1
more poshly.. as Δx tends to zero (the dx implies.. no TELLS you that)
We just picked our x1 from anywhere .. its just an x so ..
dy/dx=2x

It gets even better. e^x is out of this world.