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/sci/ - Science & Math

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15842669 No.15842669 [Reply] [Original]

what the actual fuck is differentiation? i can't wrap my head around how it works and i need to understand it for my upcoming exam please help

>> No.15842672

consider a straight line.
you (should) know that it can be represented as y=mx+b, where m is the slope.
Now consider a curved line. You can't apply the concept of a slope to the curve itself. So, instead, you pick a point on the curve and consider a very, very tiny region around that curve, tiny enough that the curve is "basically straight" there. Then you find the slope of the curve at that point.
Differentiating a function is just doing this over and over again at different points, and the derivative tells you how much the slope changes at a given point.

>> No.15842682

The basic idea is that "zooming in" at a point P will make the curve look more and more like a straight line. So at "infinite zoom" you would not see a curve anymore, but an actual straight like. The slope of that like is called the differential at P.
Of course there could be a curve that doesn't look more an more like a straight line when zooming in, like fractals and shit. These "bad curves" are called "not differentiable". You can safely ignore them when learning the basics of differentials.

>> No.15842686

One of my favorite equations for understanding derivatives is
[eqn]f(x+\Delta x)\approx f(x)+\Delta xf'(x)[/eqn]
which gets better and better as [math]\Delta x\to 0[/math]

>> No.15842693

The rate of change

Consider the equation x^2.
Plug in 1, 2, 3, 4, 5.

1 -> 1
2 -> 4
3 -> 9
4 -> 16

1 - 0 = 1
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7

The rate of change in the differences of the products are 2. Therefore the derivative, the rate of change, is 2.

d/dx (x^2) = 2x

>> No.15842715

Slope basically, which is basically rate of change of y wrto x

>> No.15843803

>>15842669
what the fuck is that pig

>> No.15843842

>>15842669
the derivative of the function is a function showing the instantaneous slope of that curve at any given point, given that it is continuous. the derivative of y = x^2 is y = 2x, so y = 2x shows the increase in the slope of y = x^2. the power rule is the simplest for finding derivatives, d/dx (notation for derivative) x^n = nx^n-1. so to differentiate this way, you take the power (sometimes the power is 1) and put it out front of the variable as a coefficient, then subtract one from the power. there's also a limit definition of derivatives that you can look into but won't fit in this post. the opposite operation of differentiation is integration, which you'll be doing if you take calc II

>> No.15844017

homework thread

>> No.15844032

>>15842669
More importantly, what is differential froms?!

>> No.15845216
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15845216

>>15842669
It's the slope of the slope.
Consider y = mx + b.
The slope is m, no matter where you are on that line the slope will always be the same.
Now consider y = x^2.
The slope is 2x at any given point. You can see the line get progressively steeper as it approaches infinity. This is because the slope is increasing as x increases.
This applies so on to more complex functions. The best real world example I can think of is velocity and acceleration. Velocity is the differentiation of acceleration, and acceleration is the change in velocity.

>> No.15845612

>>15844032
A differential form is simply an anti-symmetric (0,n)-tensor field on a manifold.

>> No.15845615

>>15843803
His name is Sam Hyde. He's a human now.

>> No.15846086

>>15842669
play this game and then you will get it
sinerider.com

>> No.15846092
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15846092

>>15843842
>the instantaneous slope of that curve at any given point, given that it is continuous
>Actually thinking that continuity necessarily implies differentiability

>> No.15846112
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15846112

how to tell Experimental questions from real questions? i don't want to waste time on them.

https://www.prometric.com/about-us/about-prometric

Experimental questions. Your examination includes 15 experimental questions that will not be scored. They are distributed throughout the examination and will not be identified as such. These are used to gather statistical information on the questions before they are added to the examination as scored items. These experimental questions will not be counted for or against you in your final examination score.

>> No.15846163

>>15842669
>differentiation
a work around the fact that primitive human math isn't able to deal with anything else than a - b relations. They hide the fact in complicated wording and formulas (usually they put some dumb history on top) were in reality it is a dumb crutch made out of no-nexisting lines.

Thank me later

>> No.15846176

>>15842669
how a function changes at a single point.

>> No.15846210

>>15842669
use the limit definition

>> No.15846435

/sci/ is actually the retard board confirmed

>> No.15846476

>>15842672
> derivative tells you how much the slope changes at a given point
that would be the second derivative

>> No.15846487

>>15842669
just inherently divide by 0 bro its cool don't worry about it
>math
lmao, what a load of CRAP

>> No.15846538

>>15846487
filtered

>> No.15846557

>>15845612
I don't know what a tensor is.

>> No.15847391

>>15846557
>>15846557
An (n,m) tensor is merely a multilinear function on a vector space which takes n covectors and m vectors as input and returns a scalar

>> No.15847528

>>15842669
Underage

>> No.15848979

>>15842669
How a small change in a quantity x affects the quantity y

>> No.15849003

>>15842669
instantaneous rate of change of a function with respect to its variable
speed is a the rate of change of distance with respect to time, ie how fast you’re going is determined by how much distance you cover in a given time. The speed at every point in time is calculated by recording the distance covered in the small time period around the point of interest. So if you want to know your speed at 10:34, you would record the distance traveled between 10:33 and 10:35 and divide it by 2 minutes. The finer the time interval, the better is your speed estimate. This refinement is done ad infinitum in case of a true derivative.