The Fundamental Principles of
Integral and Differential Calculus

'm making this page because the other pages I've seen on the subject just aren't fundamental enough. They say too much yet convey little ... too much jargon and not enough conceptualization. They start out with one sentence then load you up with parenthesis, obfuscating subscripts and superscripts, numbers and consonants mixed all up together and hosts of other flic and flam that looks like fireworks, smoke and mirrors ... in type. You're bogged down from the get-go trying to follow what ought to be simple reasoning ... a steak with one hundred sauces and condiments on top, sides and bottom. Sheesh! What a mess!

Hope I can fill the gap somewhat.

So Let's Start Here ...

There is an element of dimensional analysis to be considered first. If we add ten grapefruit to seven peaches we get ... what? Something like mango juice? As they say, you can't add apples and oranges, i.e. you can add 10 to 7 to get 17 but the dimensions (grapefruit and peaches) are not compatible. To get meaningful compatibility you need to multiply or divide. This is the way physical reality is constructed. You have ratios with division (length / time = velocity), or new meaningful entities with multiplication (length x width = area).

Multiplication and division "marry" parameters together to get a new "species" ... very often viable offspring ... not just a monkey with one wing and one arm and two left flippers flapping and squawking like a headless chicken.

We are concerned here mainly with physical relationships by which parameters, defined as functions of others, operate in consistent mathematical fashion.


A derivative is then simply (in terms of Cartesian coordinates)

... y divided by x ...
which yields a ratio between y and x. That is, it's the slope of a line.

But you can't draw any specific slope at a point (x,y) because there isn't any way to choose from amongst the possibilities. For choosing, we need two points separated by some distance.

Now, above, the slope is 1/2. That is y over x equals 1/2. If this is a straight line, the slope at any point on the line is the same 1/2.

The ratio y/x in trigonometry is called the tangent (side opposite over side adjacent). You can see why they call a line touching an arc at one point, a tangent. If you graph the thing, that line is simply a "y over x" deal ... a constant ratio ... the tangent ... the slope of the arc at that point.

So, the whole thing of differentiation is to find the slope of a curve at a specific point ... or ... the rate (ratio) that y is changing with respect to x at that point where the "tangent" is touching.

Actually, the whole thing is that you understand the problem and somebody else (a mathematician) figures out the general answer and you just "look it up". But we'll get to that later.


Integration is basically multiplying .

.. x times y ...
to get ... an area. But we want the area under a curve ... not the area of a square box (that's too easy to warrant a new branch of mathematics). A box area is what you would get if you just multiply y times x. What we have to do is multiply y times x in some such way so as to calculate the area under a curve (or any section of that area).

In fact, with either differentiation or integration, you don't get a specific answer. Rather, you get another equation from which you obtain the specific answer you want. So instead of getting say, Answer = 258, you get Answer = x2+2 and you put in your "x" which here would be 16 and come up with 258.

Inverse Functions

Differentiation and Integration are inverse functions. One undoes the other. Thus, if you multiply x times y and get the "new equation" ... you have the old equation's integral. If you take the new equation which is "xy" (alias "the new y") and divide by x ... xy / x ... you get the old equation back ... which is "the old y". Get it?

So as bizarre as it may seem intuitively, the slope of a line tangent to a curve is inversely related to the area under the curve.

If your equation is x2 ... it's integral is ... 1/3 x3 ... and the derivative of the new equation (1/3 x3) is ... x2.

I mean of course that y = 1/3 x3. Or, in jargon terms (which I personally despise)

y = F(x) = 1/3 x3 ... y is a Function of x.

It's related to x because it contains the "x" variable. See?

I don't like it when they use letters to mean something other than they normally mean. F(x) looks like F times x in normal notation. Why do they do this? See my tome on the "House of Mathematics".
So, you can take the derivative of the derivative of the derivative of the derivative (1st, 2nd, 3rd, 4th derivatives ...) as long as they give an answer. If you keep taking derivatives you generally simplify till there is nothing left, i.e. the process runs out.

And you can do the same with integration. 1st, 2nd, 3rd, 4th ... integrals.

You work the above 3rd integral from the inside out, i.e. do the one shown first ... then whatever you get ... do it again ... then that result once more.
Going up is integration. Going down undoes the integrals and they are derivatives. The 3rd integral here is the 4th integral's derivative.

Get it?

If you take the integral of y with respect to x, you don't have to do anything else to get the derivative of that integral. It's the original equation you started with.

Every equation is another equation's integral and yet another equation's derivative.
Provided that you haven't "bottomed out" on the derivatives, that is. (See below)

All right ... on the the actual business

Everything has its little gimmick that is the most fundamental thing about it. It's the key that unlocks the door to the treasure room.

The "key" here was ground out mainly by Newton and independently by Liebniz. You stick it in the lock and hoards of mathematicians can make a living and engineers can build really big stuff that doesn't fall down and send a man to Mars (soon! Godammit) ... and astronomers can calculate the the guts out of a "black hole".

The Fundamental Gimmick

To find the slope of a curve at a given point, you want to find the slope of the line segment connecting two points on that curve as those points are brought closer and closer together. The closer they get, the more closely the slope of a line connecting them comes to the "true" value. That value is the limit as the segment approaches "0" length.

What we really want here is the ratio of y to x in that line segment when it approaches a limit. What is that limit?

an infinitesimal increment added to y
divided by
the corresponding infinitesimal increment added to x
Note: We don't care about the lenght of the line. That will of course be zero at the limit. We want the slope (y/x) of that line at the limit which is non-zero.
Well, the increment of x we need is ... (ix) = (x+ ix) - x ... obviously (x plus the increment then minus the original x).
I'm using i for increment instead of the greek letter delta (a triangle) for browser reasons.
And that of iy is (in terms of x ... remember y is a function of x like, hmmmm ... let's use x3)

So ... the increment of y will be equal to ...

iy = (x + ix)3 - x3

Here we add the increment to x and cube it ... then subtract the original part (without the increment) ... leaving just the increment itself (the little "y" component of the diminishing line segment).

Expanding the above we get

iy = [x3 + 3x2(ix) + 3x(ix)2 + (ix)3] - x3

Therefore ...

             3       2               2        3      3
iy          x   +  3x (ix)  +  3x(ix)  +  (ix)  -   x
___  =    _____________________________________________

ix                         (x + ix) - x


                   2               2        3
iy               3x (ix)  +  3x(ix)  +  (ix)
___    =     __________________________________

ix                            (ix)


iy               2                    2
___    =       3x   +  3x(ix)  +  (ix)


    so ..........

iy             dy                  2
___   =        ___       =       3x

ix             dx

The other terms are just zeroed out when (ix) becomes zero (at the limit).
dy/dx is Leibniz' notation for iy/ix (which I also object to).
It really ought to be something like this.
But that's another story.

Thus, the slope of any point on the curve y = x3, can be reckoned with the new equation
y = 3x2
which is the derivative of y = x3 .

So, what is the slope of the curve y = x3 when x = 5 ?

Answer: 75 ....... [that's 3 (52) ] ... and that's pretty steep. At that point, for every 1 increment of x to the right, y goes up 75 increments.

Note: When you differentiate you decrement the degree of the equation, e.g. 3rd degree to 2nd degree. When you integrate, you increment the degree of the equation as from 2nd to 3rd. That's why you can run out of derivatives.
x3 to x2 to x1 to ......... x0 = 1 ....... and it's hasta la vista variable.
No variable = no F(x).

Now remember,

the integral of 3x2 is x3
and the derivative of x3 is 3x2

That's called

"The Fundamental Theorem of Calculus"

So what's the "gimmick" of integration?

There is no gimmick.

If you want the integral of a given equation, you know that you have the derivative of the desired equation in hand ... because the original equation is ... related to ... it's integral.

Since you already have the derivative, you just have to reason backwards and say, "What equation will give me this equation as its derivative?".

Integration is induction.
Differentiation is deduction.

Therefore, integration is much more difficult. So, when a mathematician finds a difficult integral (or a difficult derivative for that matter) ... they put them in generic tables ... in books ... so others can "look them up" and not have to do the hard work again.

Now, when you want to integrate a function you have a program like MathCad which will look into its database (something like Windows installed driver base) and find the right form for your integral ... then stick in your specific constants ... and ... voila! ... there it is!

Remember, computers can't do induction. They just do deduction. Now, if you ask for a derivative, it can deduce that to some extent since it's deductive.

Also, the "antiderivative" is just the integral ... but divorced from any geometric base. You can think in purely abstract mathematical terms here. It's ... OK.

What should you learn?

If you are studying the subject because you like it ... or ... because you are going to be a research mathematician, you should put as much in your head as possible.

If you want to be an engineer, put as much in your head (formulas) as you can readily utilize. This means that if you can put a formula to use from memory faster than you can "look it up" ... memorize it!

But if you're just putzing around with math ... understand the thing ... remember where to look for that type of problem ... and go on to other things ... like art or the hot tub ... just whatever. I have never been able to think of any reason to make one's mind into an encyclopedia. Philosophically, one should increase the level of one's general understanding along with some rote memorization ... but not 95% rote memorization and only 5% understanding. My mind is not a 100 gigbyte hard drive for miscellaneous unused information storage.

I will never win a trivia contest.

PS - I strongly recommend "Quick Calculus" by Daniel Kleppner and Norman Ramsey if you are interested and presently know next to nothing. There is of course much more to the subject than I could ever touch on.


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