What does this shit mean?

The equation (f(x+h)-f(x))/h as h tends to zero.

Linear approximations.

robjlow.blogspot.com.tr/2017/07/the-right-way-to-think-about-derivatives.html

lol i remember asking this very same question when i was learning calculus.

wrt, or "with respect to," is really just naming the variables. like if

f(x) = y = x^2

then dy/dx would just be

= 2x

so to specify both variables in that function (x, y), we would say we just took the derivative of y with respect to x.

just like if you name your function

z = a^2

then dz/da

= 2a

you just took the derivative of z with respect to a.

this is particularly evident when you consider a function with more than 1 variable, such as

y = 2x + 3w

you now have 2 variables (x,w) to choose to differentiate. so you could take dy/dx or dy/dw, where

dy/dx = 2

and

dy/dw = 3

Thanks friend

any other questions?

The notation originates from Leibniz's calculus. Newton just used dots to denote differentiation. So df/dx=f'(x).

The idea is that dy is an infinitely small change in y when you vary x by an infinitely small amount dx. Infinitely small means that (dx)^2=0.

An example should clear things up. Let y=x^2. Then y+dy = (x+dx)^2 = x^2 + 2x dx + (dx)^2. Now since dx is very small (dx)^2=0 giving y+dy=x^2+2xdx. Subtract y=x^2 from both sides to get dy=2xdx. So the ratio of dy(change in y) to dx(change in x) is dy/dx=2x. This result is interpreted as the slope of y=x^2 as a function of x. So dy/dx at x=2 is 4, which means right around x=2, varying x by a causes y to increase by 2a.

Another example y=sinx. y+dy=sin(x+dx)=sin(x)cos(dx)+cos(x)sin(dx). For small angles cos(x)=1 and sin(x)=x so y+dy=sin(x)+cos(x)dx. dy=cos(x)dx. dy/dx=cos(x).

d/dx on anything gives d(anything)/dx. dy/dx is the derivative of y with respect to x. So if y=ax then dy/dx=a and dy/da=x (assuming that x is not a function of a, otherwise we use the product rule to get dy/da=x+a dx/da).

This calculus is not rigorous. This is because infinitely small quantities are not clearly defined. Most modern sources simply use the idea of limits instead of trying to extend the number system.

Don't bother with these explanations. They're awful.

Layman explanation: d is just a tiny difference ("differential"). You're probably familiar with delta x (x final - x initial) which is similar, but denotes a more noticeable, algebraic change. For the slope of a line, you measure rise over run, which is just delta y over delta x, right? But when you no longer have a line, you have no good notion of slope for the entire thing. So, you give up with that and instead give the value of a line AT A SINGLE POINT. You just take the little tiny rise over the little tiny run, and thus you have the graphical idea of the derivative.

Honestly, I'm amazed at how many people can only talk about things like math in terms of either machinery or properties...it's like 99% of people here never even tried to think. No wonder society thinks science and math are filled with autists.

[math]dy:the\; change\; in\; y \\ - \;: over \\dx :the\; change\; in\; x[/math]

>In layman's terms
There are no lame-man's terms for the Calculus,
bcoz it is not for the double-digit IQ mainstream.
>Please no smartass/dickhead comments.
>Thanks in advance.
Pls excuse my comments, fuck you in advance.

the d stands for delta, dy/dx is the same as Δy/Δx, which just means the change in y over the change in x

Of course, we can make this change in y over change in x smaller and smaller, so small that it becomes a point. If we let the difference between the two points become zero then Δy/Δx becomes dy/dx. Thats how calculus started, you take two points on a function, and let the difference between them approach zero- which lets us find the slope, the rate of change, throughout the function

To do this, we calculate it directly from that Δy/Δx, or dy/dx. The change in y is the difference between the function and the closest point we can make, which is f(x+n) - f(x). And the change of x is just x+n - x, or n. From that we get [f(x+n)-f(x)]/n

and we made this all with the distinction that we're letting n approach zero, so we take the limit as n approaches zero

dy/dx is important because its the rate of change, the derivative of a function is the rate of change for it. The derivative of distance is velocity, because velocity is the change in distance over the change in time, and the derivative of velocity is acceleration because acceleration is the change in velocity over time.

I dont remember how newton notated it though

Suppose you have a function y(x) — often just written y. If you want to find its gradient, then youll differentiate it. To do this you'll put your function y into the derivative function. The derivative function takes other functions as inputs and then outputs their derivative. This derivative function will often look like [math]\frac{\mathrm{d}}{\mathrm{d}x}()[/math], [math]D_x ()[/math], [math]\frac{\mathrm{d}()}{\mathrm{d}x}[/math] where your function will be between the parentheses. Other notation may involve affixing something to your function, such as y'(x) or [math]\dot{y}[/math]. The [math]D_x ()[/math], [math]\frac{\mathrm{d}()}{\mathrm{d}x}[/math] uses an x to show what variable you're differentiating with respect to. when differentiating with respect to x youre basically differentiating the x terms as usual and treating other variables as constants. in the other notations it is implicit what you are differentiating w/r/t.
at this point don't be confused into thinking that dy and dx are separate elements constituting a fraction when in the form dy/dx, or that dy/dx somehow means a small change in y over a small change in x. that's wrong; you should treat d/dx as a whole*. the bar between the d and dx does not represent a fraction and there for historical reasons.

for integrals we use the notation [math]\displaystyle \int \square \, \mathrm{d}x[/math] to integrate whatever might be contained within [math]\square[/math] with respect to x. that's pretty much it. the dx, at this point, is meaningless on its own and is just part of the notation. historically the dx probably comes from the delta x in [math]\displaystyle \int_a^b f(x)\,\mathrm{d}x =\lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x[/math]

*although, by coincidence, it can serve to make integration by substitution and differential equations easier when the dy and dx can be separated

dy/dx is a ratio of infinitesimals dy and dx provided the function is continuous.
dy=f'(x)dx because of how linear approximations work, dy is a change in the linear approximation, and dx is the increment, then dy/dx = f'(x).
Ignore anybody who tells you otherwise.

most wrong post itt

>Disclaimer: written "in layman's terms", as OP asked

Think of the "d" as "delta", "distance" or "difference".

dy / dx => "height difference / width difference" => "how much units of y do we have for each unit of x"

It is more complicated, because the units are kinda "infinitely small", but that's the genaral reasoning.

Therefore an itegral is more or less just a sum+limit of "tiny parts" which form an area. Each of this parts is infinitely small (dx) and the height is f(x).

So you can read: "integral from 0 to 1 of [x * dx]" => LIMIT FROM n=0 TO INFINITY OF (SUM OF x(n)) => x(0) + x(1) + ... + x(n)

Each "x(i)" is a really really thin area here, so to say.

Ok everyone is getting way too complicated with their explanations.

All it is is how fast the function is changing. It's like a speed, pretty much.

Look at a function like x^3. It shoots off pretty fast, doesn't it? You can look at the graph with any kind of viewing window or ANY scale and you will still see it shooting off fast. There is just something inherent about that shape that it speed off.

Now look at log(x). You will see it's always increasing, it is getting larger, but it grows very slowly. You can stretch the graph and adjust the scale all you want, but you will always get this lazy curve going off to the right.

So we can see that x^3 is much faster than ln(x), so we can conclude that the derivative of the first function will be bigger than the derivative of the second.

Forgot pic.

OP, every function has a certain speed. The derivative just gives you the exact value of this speed.

It's just a fucking limit.

what about the whole function?the speed is the average of a closed set of points. this is only true of cauchy functions right?

>calculus
>not the double-digit IQ mainstream tier math
Found the engineer

*quotient
[eqn]\frac{\lim_{h\to 0}f(x+h)-f(x)}{\lim_{h\to 0} h}[/eqn]

kek too true

>In layman's terms

It's a ratio of two variables that describes how one changes when the other changes. It's abstract at first, but then you find out it has very important implications in physics and mechanical engineering.

d represents infinitesimal increments i.e. limits
so dy/dx is an infintesimal change in y with respect to x, much like delta y / delta x (slope) from algebra, but with limits

youtube.com/watch?v=rjLJIVoQxz4

so what does d stand for? is it a function?
dont give me any of that it's an infinitesimal shit

it's a operator

Newton's D

A FUCKING limit nontheless

So basically "with respect to" means when y=x in an equation. If I just disregarded the "respect to" thing then x could be anything becuase there is no "law" stating it is bound by another variable?

it's a small "d" because he never got laid.

Right but you can take the limit of numerous things. In the context of this thread it is arguably more informative to call it a quotient. Of course it is more accurate to say it is "the limiting value of a particular quotient," but i digress.

>he never got laid
Oh yeah? He ravaged Lebniz' ass pretty bad.

(((anglo)))

This is wrong.
This is the answer you're looking for.

Watch this series:

youtube.com/watch?v=WUvTyaaNkzM

Seconding this

Not really, because you might be working with a multi variable equation.

For example, the partial derivative of the function f(x,y,z)=2x^2-y+4

w.r.t. x: 4x
w.r.t. y: -1
w.r.t. z: 0

Its a slope for a linear equation not just a quotient if you want to be rigormortous about it.

I find you understand things better when you can put them in your own words. When i took calc 1 i had to do just that when first introduced to the derivative. Here is what i found scribbled in my notes 6 years ago when i took the course.

Given a function y = f (x) (so y is a function of x), the derivative of f is
f ' (x) = dy/dx.
That is, the derivative is itself also a function of x; it is a function that returns the instantaneous rate if change of f at any given x.

The symbol d/dx is an operator (similar to a square root sign, or log (x)). It takes in functions (like y = f(x)), and returns the derivative function of said function.

So if
[math] y = f (x) = x^2 [/math]

Applying the operator (i.e. taking the derivative) gives us

[math] \frac{d}{dx} [f (x)] = \frac{d}{dx} [x^{2}] = 2x = \frac{dy}{dx}= f'(x) [/math]

change of y in terms of change in x

This really, Most can't grasp, especially if they have no force behind them to elevate understandings.

>The more understand, the easier it is for others to learn

they're differentials which go away using limits

>simple
>to the point
I like you.

>What is a derivative
what has happened to this board

>how did newton notate this stuff

he stole the answers off of Leibniz

top part of the equation means do it yourself essentially The bottom half is missing an M .

Infinitely small change of something

...

It's useful to approximate the rate of change of a function in respect to one of its variables. You probably already know the formula for estimating it. A derivative is just the rate of change when x_2 - x_1 is infinitesimal, i.e. approaching zero.

Think of it as a 0/0 limit.