## Mathematical Limerick

### Limerick to start us off :)

$$ \cfrac{12+20+144+3\sqrt{4}}{7} + (5\times11)=9^{2}+0 $$

A dozen, a gross, and a score. Plus three times the square root of four. Divided by seven. Plus five times eleven. Equals nine squared and not a bit more.

## Derivative by First Principle

At first glance, these equations can be a bit hectic (at least it was for me). There is the use of limits and in the end it all ends up being pretty long. But! I would consider it kinda essential to understand how these work in order to understand how all the formulas work for derivitives.

And in the end, I think they are kinda neat :)

So, let us start with what does the equation look like?

$$ f'(x)= \displaystyle\lim_{h\to 0} \cfrac{f(x+h)-f(x)}{h} $$

Amazing, beautiful, gorgeous. Some poeple will use the Greek Letter Epsilon, ε, instead of 'h', but cause I don't wanna get the code for epsilon's symbol each time, I'm gonna use 'h'. Alright, next step is picking an equation and then substituting it into our lovely formula. For simplicity, I am just gonna use a basic little function.

$$ \begin{split} f(x) &=2x+2 \\ f'(x) &= \displaystyle\lim_{h\to 0} \cfrac{2(x+h)+2-(2x+2)}{h} \\ &= \displaystyle\lim_{h\to 0} \cfrac{2x+2h+2-2x-2}{h} \\ &= \displaystyle\lim_{h\to 0} \cfrac{2h}{h} \\ &=2 \end{split} $$

This also works if our function has any indicies in it, as demonstrated below.

$$ \begin{split} f(x) &=2x^{2}+2 \\ f'(x) &= \displaystyle\lim_{h\to 0} \cfrac{2(x+h)^{2}+2-(2x^{2}+2)}{h} \\ &= \displaystyle\lim_{h\to 0} \cfrac{2(x^{2}+2xh+h^{2})+2-2x^{2}-2}{h} \\ &= \displaystyle\lim_{h\to 0} \cfrac{2x^{2}+4xh+2h^{2}-2x^{2}}{h} \\ &= \displaystyle\lim_{h\to 0} \cfrac{4xh+2h^2}{h} \\ &= \displaystyle\lim_{h\to 0} 4x+2h \\ &= 4x \end{split} $$

And that's what gives us all the rules for derivatives :) The more times different equations are done this way, a pattern appears as to what the function originally looked like and what it ends up looking like. In the examples I used, the first one is a simple, no-powers function, where two things interesting happened. The constant of '2' disappeared and so did the variable 'x'. By comparing this to the second function that **does** have a power, the variable of 'x' stays behind while once again the constant goes away. This was one of the ways that I was shown how the derivative rules make sense, by putting a ton of different functions into this here little formula.

And, that wraps up all that I have to say on derivatives by first principle :)

Something that I'll add actually, is that from my understanding when you are thinking about this type of derivatives it highlights an important aspect. Which is that it is a averagae of a lot of little distances along the line.

As in, each time epsilon is added, that is a new tiny, eeny weeny, segment. Then to find the derivitive, you take what the distances are and divide by what your adding (epsilon) (or h).

## ♥Derivatives♥

I like derivatives. I like order of first principles. I like the squiggles on the graph and making them less squiggly.

Alright, lets get into it. The nitty gritty. The space where I spend far too long going over why order of first principles is amazing and deserves respect while ignoring limits at all costs

In the first principle section (above) I went over most of the aspects of why it is good for the basics and that it reveals the different patterns associated with derivative rules. One of the big reasons why I at least like it, is because it sets up a good foundation for all the rules. To go over each and every rule by putting all the different functions through the first principle stuff gets very long and I consider it unnecessary for that purpose. But! I reckon, if the first principle stuff makes sense for the basic functions, the pattern learnt carries over to all the other rules making it a bit less overwhelming.

## Basic Formulas

So, let us start with the summary of just a basic, nothing complicated function.

Just a note for the unfamiliar, the dash "$ ' $" or $ \cfrac{dy}{dx} $ signals a derivative

$$ \begin{split} y &=f(x)^{n} \\ y' &= nf'(x)[f(x)]^{n-1}\\ \end{split} $$

Above is the simple formula, used for the average function. It can be applied to each individual element of the function. To see an example, see the derivatives by first principle section (above), and observe how the function $ f(x)=2x^{2}+2 $ had this particular formula pattern applied to both $ 2x^{2} $ and $ 2 $.

$$ \begin{split} y &=uv \\ y' &= v'u + u'v \\ \end{split} $$

This one is referred to as the product rule, where two seperate, distince function will be multiplied together. E.g. it would be used for: $ (2x+2)(4x^{2}-7) $, though I would probably recommend simplification as the first point of call, I just couldn't think quickly enough of a better example.

$$ \begin{split} y &=\cfrac{u}{v} \\ y' &= \cfrac{v'u + u'v}{v^{2}} \\ \end{split} $$

And that's the quotient rule :) Opposite of product, used for when two functions are being divided, definitely beneficial for fraction based stuff. Something to note though, is if you have an equation like: $\cfrac{2x^{2}+2}{2x^{-2}}$ then becuase the denominator is a negative, by taking the inverse and bringing it to the top, the simplifaction process becomes a bit easier. And that way you can use a more basic formula rather than the quotient rule. This can work even if the denominator isn't negetive since it can work as a negetive anyway.

An example of what I mean by that is just that say you have an equation that looks a bit funky and you dont really wanna use the quotient rule. Lets say it is $\cfrac{3x^3+4}{x^{-2}}$ you can rearrange it into $x^{2}(3x^{3}+4)$ and use the product rule instead :)

Some other important formulas are:

$$ \begin{equation} \begin{split} y &= \ln f(x) \\ y' &= \cfrac{f'(x)}{f(x)} \\ \end{split} \end{equation} $$

$$ \begin{equation} \begin{split} y &= e^{f(x)} \\ y' &= f'(x)e^{f(x)} \\ \end{split} \end{equation} $$

$$ \begin{equation} \begin{split} y &= \log_af(x) \\ y' &= (\ln a)f'(x)a^{f(x)} \\ \end{split} \end{equation} $$

There are other formulas, most of the ones that I'm familiar with are related to trig, but I don't find it necessary to mention here.

Derivatives are my favourite type of mathematics (at least for now XD) and the biggest reason for that is that once I memorise one of the formulas, and can recognise each new function as which formula to use, it becomes incredibly easy to perform the calculations. Because at that point it becomes as simple as substitution yet there is still enough manipulation and algebra and what not that keeps me hooked and interested.

## Graphs

Alright, this is still a work in process. I have got a something of a graph, but it isn't all that I want of it, so I've got some more to do. I want to have the axis and each line/curve to be labelled which is proving to be more difficult than I thought.

**WORK IN PROGRESS!!! TRYING TO CODE SOME COOL LOOKING GRAPHS!!! BEWARE YOUR EYES!!!**

The step after this is anti-derivatives/intergrals which is just all that is the above but reversed. Though, I like intergrals less as it incorporates a few more/different aspects into the process and some of the different steps simply do not stick in my head.

## ☠Intergrals☠

## Trigonometric Identities

Alright, this one goes out to my senior math teacher from way back for putting up with me while we were needing to work with this topic. Thank you for never giving up on me even though it took me like two years for it top finally click. If it weren't for you Ms Kel-dog, I probs wouldn't have got all the other proof work and identity laws that go into Boolian or anything else for that matter.

So, lets take a look at it. We'll start with the basic stuff, sin, tan and cos and their opposites.

$$ \sin(A)=\cfrac{1}{\cosec(A)}, \cosec(A)\not=0 $$

$$ \tan(A)=\cfrac{1}{\cot(A)}, \cot(A)\not=0 $$

$$ \cos(A)=\cfrac{1}{\sec(A)}, \sec(A)\not=0 $$

Note: this part was one of the hardest parts for me tbh. But I eventually got it becuase if you look at the graphs, it is quite visually obvious that each trig function has an opposite. I'll probs add graphs later on.

And all that stuff can be reversed just by rearranging the equation so that the denominator is now the focal point.

Next, there are the slightly less obvious ones that are fairly important to remember (or at least have written down on a reference sheet) if working with this stuff.

$$ \cot(A)=\cfrac{\cos(A)}{\sin(A)}, \sin(A)\not=0 $$

It can also be swapped around for tan, since cot and tan are opposites

$$ \cos^{2}(x)+\sin^{2}(x)=1 $$

## Boolian Logic

All proofs kinda loop in on themselves and end up being the same. You cannot convince me otherwise that there is an important difference between the proof works between Boolian, triangle congruance, trig proof work, and Literally^{TM} any other type of proof work. They ALL end up being the same

Though I will give trig a small break considering that I actually think that 1+tan^{2}(θ)=sec^{2}(θ) and 1+cot^{2}(θ)=cosec^{2}(θ) is based.

## Fav Miscellenous Equations

a^{2}+b^{2}=c^{2}

$$ a^{2}+b^{2}=c^{2} $$