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.

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).

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.

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.

Note for self: Add unicode symbol for skull (or whatever is closest/best) to title and then procrastinate against finishing this section at all costs

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

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

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+tan²(θ)=sec²(θ) and 1+cot²(θ)=cosec²(θ) is based.

a²+b²=c²

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

04/11/2023

I am very excited! Its gonna be all about algebra, manipulation and computation. The professor said there were multiple aspects of last semesters class (2023, Sem 2) that would have been assisted with this knowledge but oh well, we can only progress in a linear form sadly enough.

Currenlty I am taking a unit on applied calculus :). Specifically looking at implicit function theorum, multivariable calculus, dysnamical systems, and others, with the applications being Lorenz model of the atmosphere, Lotka-Volterra model, elementary theoretical and celestial mechanics, quantum mechanics and so on.

I did modelling and discrete maths last semsester and it was really fun so I expect as much from this unit. And!!! I've got the same lecturer for all three of my math units lol. Luckily I like 'em :)

I'm also doing an intro to info tech unit, where I'm looking at html, python and other beginner stuff. Hopefully, as I learn stuff I can improve this whole website lol.

04/11/2023

Alright lads, I did it! I pushed through it and have officially completed my first year of uni! Whoop whoop!
The last math test was alright, just multivariable calculus with not really that many worded questions. Though, the assessment is a different story. I submitted that today so we will just have to wait and see how that goes. I 100% know that I got question 2 wrong. It was to do with a tank of water that initially had 30kg of salt in it, there was water entering and exiting, making it that the solution was diluting and the rate of salt leaving was slowing. For part c) it asked how much salt is remaining after an hour. Well, your boy right here, go the answer of 538kg B) I'm pretty sure it was becuase I have either made an error in interpretation of the question, becuase it was a worded question (yucky), and/or the error was made in the integration stage where I only had one constant instead of two? Not sure though, oh well!
Next semester I should be taking a class on algebra and what not, supposed to deal with methods of computation and manipulation I think, like matrices and what not. Either way I am excited! :)