Conceptually explain to a brainlet what exactly a derivative and integral are

conceptually explain to a brainlet what exactly a derivative and integral are

also explain the point of having dx within the integral

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You can think of dx just as part of symbolic description of integral showing the unknown that is integrated along, without meaning on its own, that is AFAIK how it's done in the most undergrad texts. Intuitively it's a side of a rectangle in a Riemann sum.

>also explain the point of having dx within the integral
If you have many variables then you need to know with respect to which variable you are integrating.

A derivative graphically symbolizes the slope of a tangent you can draw at any specific point on a curve. While physically it represents the instantaneous change of one quantity with respect to other.
Integration onother hand is represented as the reverse of derivative however has a completely different meaning where dx is just a symbolic representation of telling you with respect to witch variable you are integrating a given function.Graphically integration represents the total area under a graph i.e if u divide a graph in (h) number of parts where h is almost equal to infinity then integration is the complete sum of all those parts.

want me to describe them more?

>Derivative
Rate of change
>Integral
Area under the curve

so her integral must be 0 then

>Derivative
a point slope
>Integral
an instruction to sum underneath the curve
whether it's underneath or to the left depend on whether you say dx or dy etc.

The derivative is the limit of the difference between y-values divided by the limit in the difference between x-values.
The integral is the positive sum of the y-values multiplied by the limit in the difference between x-values.
dx is shorthand for the limit as x approaches 0.

>dx is shorthand for the limit as x approaches 0.

Engineers shouldn't get near math threads

Then what is it?

dx as in dy/dx is just a notation, not a fraction. Same thing in integration.

You can define differentials in differential geometry in multiple ways, but it's meaningless here without context.

Not really, skin has non-zero width.

That's dumb. It should have a meaning.

u dum
just cus you don't know the meaning doesn't mean there isn't one

It does, it comes from the idea of infinitesimals.
When calculus was developed we used the idea of "infinitely small" numbers. (unfortunately it wasn't rigorous and was overcome by limits which were rigorous but less intuitive) [btw infinitesimals were made rigorous in 1960, so working with them is now possible with some reading]

when you take dy/dx you take a fraction
it tells you how much y changes w/ respect to x
it makes sense when you think about right triangles with infinitesimal lengths

integration is an infinite sum of rectangles.
you take rectangles that get thinner and thinner until they become infinitesimal in width, which is why you can add infinite of them and get a finite number.
f(x) * dx is a formula for area where f(x) is height and dx is width

>derivative
subtraction
>integral
addition

no

So I'm right then? It is a fraction (a slope)?

yes

The image could be simpler but focus on the right triangle.
On the bottom it says dx, dy is unmarked.
dy/dx follows the form of tangent, which you may recall, is defined as:
opposite/adjacent
if dy is very large then tan is large
if dy is small then tan is small
dx is your independent variable, so it is where you have 'control'

what this means in the end is that if the derivative is large then a small movement in x will mean a large movement in y
if the derivative is very negative -> a small (positive) movement in x will mean a large (negative) movement in y
if the derivative is zero then a movement in x means no movement in y

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That is actually what dx means though...

Go back to first year calc and read up on Riemann sums

ur welcome bro

>whether it's underneath or to the left depend on whether you say dx or dy etc.
No, it doesn´t. Integrating along the y axis simply changes the axis on which the limits of integration are found - the area is always limited by a curve and two points on an axis.