Welcome to Gaia! ::

I'm working on making a roleplaying book called SCIENCE and plan on using lots of material from the classics, as well as compiling the best possible list of instructions for specific categories including:

Mathematics
Physics
Chemistry
Biology

I'm still trying to figure out the roots of the Natural Philosophy, at least in a shape. The classical liberal arts pattern includes some sciences too. I'm also looking at Trivium and modern Liberal Arts college models.

I do not feel math is being taught properly today, the most legitimate examples being our understanding of


logs
square roots
trigonometry
limits
derivatives
division
integrals

anyone who tells you that you must know how to derive before being ready for integrals is a liar. Anyone who says you must understand limits before derivatives is also a liar.

for example, in text books, you see something to the effect of f'(x) x^2 = 2x and then claim that explains the principle.

It doesn't.

Following that logic f'(x) x^5 = 5x. This is where the teachers play the "lets mess with people's heads" game.


It is much more effective to create an example using 3's and 7's.

for example, to derive x^3 we end up with 3x^2
obviously we need to know where the 2 came from. if we say the derivative of x^7 is 7x^6, and compare it,

x^3 arrow 3x^2
x^7 arrow 7x^6

we can see a pattern: the power becomes a multiplier of x and then drops by 1.

making a short chart also helps people get the basic idea:

x^1 --> 1x^0
x^2 --> 2x^1
x^3 --> 3x^2
x^4 --> 4x^3
x^5 --> 5x^4

we can then look at two special cases based on rules we are thought to know, but might have forgotten:

Special Case 001
x^2 --> 2x^1
2x^1, is the same as 2x. When x is raised to the first power, it is not necessary to note that it is raised to the first power, but not doing so when learning derivatives opens the doorway to confusion.

Special Case 002
x^1 --> 1x^0
When any number, or even a variable like n, x, y, or z is raised to the zero power, it turns into 1. This is because raising the number to the negative first power (x^-1) is the same as 1/x. So we are really talking about fractions.

x^0 is the same as X/X, while X^1 is the same as X/1, and X^2 is the same as X*X/1.

Suppose X were 5.

x/x then equals 5/5, which really is 1.

so Power Progression Might Look something like this:
let x = 5...

x^2 = x*x/1 = 5*5/1 = 25
x^1 = x/1 = 5/1 = 5
x^0 = x/x = 5/5 = 1
x^-1 = 1/x = 1/5 = 1/5
x^-2 = 1/x*x = 1/5*5 = 1/25

If a student doesn't remember this, Derivatives can be a nightmare.

So if x^1 is derived, then we first multiply the x by the power.

1 times x is still x...

then we subtract 1 from the power.

1-1 = 0.
X^0 = 1.


Important!
multiple derivatives will eventually turn into zero.

(x^1)' = 1x^0; while (x^1)'' = (x^0)' = 0x^-1

which is the same as 1/x * 0, AKA, zero.

I found a book called "Technical Mathematics with Calculus" I've already ordered a copy off amazon and will be checking out how it approaches things like natural log and trig.