Difference between Differential And Differentiation

[HemantChauhan07]

I've tried goggling it but couldn't understand it properly. What is the difference between Differential and Differentiation. Please anyone explain it with Same example.

[swansont]

Do you know what a derivative is?

[studiot]

16 hours ago, HemantChauhan07 said:

I've tried goggling it but couldn't understand it properly. What is the difference between Differential and Differentiation. Please anyone explain it with Same example.

Please tell us more when you reply as more detail makes it easier to help.

 

Differentiation is a process.

A differential is the result of that process.

(That is differential as a noun)

But differential as an adjective refers to the difference between two values eg the differential pressure between inside and outside = (inside pressure - outside pressure)

 

But there is considerably more required to understand things.

Why is this posted in homework help ?

Is this really homework or are you asking about mathematical theory, in which case you will attract more attention and answers if you placed it in the mathematics section.

 

[HemantChauhan07]

8 hours ago, studiot said:

Please tell us more when you reply as more detail makes it easier to help.

 

Differentiation is a process.

A differential is the result of that process.

(That is differential as a noun)

But differential as an adjective refers to the difference between two values eg the differential pressure between inside and outside = (inside pressure - outside pressure)

 

But there is considerably more required to understand things.

Why is this posted in homework help ?

Is this really homework or are you asking about mathematical theory, in which case you will attract more attention and answers if you placed it in the mathematics section.

 

No Actually, I was just studying my school books and I find myself always confused between Differential and Differentiation. Currently I'm in class 10th after few days I'll be in 11th. I just thought that I should ask this in homework section But I'll remember next time.  Like you see what I'm studying.

 

 

I'm struggling in this. 

Edit: Currently I'm able to get some pieces. So no worries 👍 😉 

Edited November 18, 2023 by HemantChauhan07

[guidoLamoto]

Short answer-- 

Differentiation is a process. Eg-- You may be asked to differentiate the equation y = x^2 + 8.  You would find  dy = 2xdx. Re-arrange it to get 1 = 2x dx/dy. That's a differential equation, because it contains a differential....dx/dy is a differential.... A differential is a number. Diffeetiation is a process.

[studiot]

1 hour ago, guidoLamoto said:

dx/dy is a differential.... A differential is a number.

No that part is not correct.

 

Each of dy and dx are called differentials.

dy/dx is one way of representing the derived function also called the derivative. It is a function, not a number.

 

Being pernickety is not so important in basic calculus but becomes important in multi dimensional calculus.

[Alysdexic]

Differentials are small shifts. Differentiation finds these shifts. One's the part, the other's the finding.

[studiot]

7 hours ago, Alysdexic said:

Differentials are small shifts. Differentiation finds these shifts. One's the part, the other's the finding.

Why small?  That is not always so.

[Alysdexic]

18 hours ago, studiot said:

Why small?  That is not always so.

Infinitesimals: fundamental to calculus; essential for understanding immediate rates of change and precise calculation of derivatives. Once established, it measures change rate at any scale. Why don't you read this?

[studiot]

6 hours ago, Alysdexic said:

Infinitesimals: fundamental to calculus; essential for understanding immediate rates of change and precise calculation of derivatives. Once established, it measures change rate at any scale. Why don't you read this?

You really should get out and about more.

'differential' is both a noun and an adjective much used in technical subjects and generally some sort of difference, which can be very large.

This has little or nothing to do with infinitesimals.

 

Perhaps you should look in the fields of automotive engineering, medicine, hydroelectric engineering, clean room engineering and many more besides.

 

[guidoLamoto]

On 12/3/2023 at 5:18 AM, studiot said:

No that part is not correct.

 

Each of dy and dx are called differentials.

dy/dx is one way of representing the derived function also called the derivative. It is a function, not a number.

 

Being pernickety is not so important in basic calculus but becomes important in multi dimensional calculus.

OK, here's where we start confusing the kid in the OP--

You're right- dx and dy are differentials. A differential (called an infinitesimal by Newton originally) is the value of x when x = a - b and a and b are brought closer and closer together, x approaching zero "in the limit."

dx/dy is a differential expression and could be called a derivative in the specific case where y = x + c, but that doesn't help much in defining the term. In my example above, 2x dx is the derivative of y. When you differetiate a function, you get a derivative...When you integrate the derivative, you get the function back.

Newton was an amazing guy. He invented calculus as a 19 y/o while on hiatus from Cambridge during The Plague. How many of us did anything even remotely as ambitious while shut down during CoViD? I straightened out my underwear drawer. 

[studiot]

21 minutes ago, guidoLamoto said:

OK, here's where we start confusing the kid in the OP--

You're right- dx and dy are differentials. A differential (called an infinitesimal by Newton originally) is the value of x when x = a - b and a and b are brought closer and closer together, x approaching zero "in the limit."

dx/dy is a differential expression and could be called a derivative in the specific case where y = x + c, but that doesn't help much in defining the term. In my example above, 2x dx is the derivative of y. When you differetiate a function, you get a derivative...When you integrate the derivative, you get the function back.

Newton was an amazing guy. He invented calculus as a 19 y/o while on hiatus from Cambridge during The Plague. How many of us did anything even remotely as ambitious while shut down during CoViD? I straightened out my underwear drawer. 

'the kid' doesn't appear to have come back or I would have refrained from commenting on your post so as not to confuse him.

But we can certainly expand the discussion if you like.

For your information Newton originally invented the calculus of finite differences, before going on to the differential calculus of a single continuous variable.

For my money, he was also the greatest genius that ever lived because not only did he largely invent mathematical physics he had to invent the mathematics as well.

Others who came after always then largely had the benefit of sufficient mathematics to work with.

[guidoLamoto]

We agree about Newton's place in history, but it's difficult to fault Leonardo- scientist, artist and futurist.

If we were chosng up sides for a stick ball game in the streets and Mickey Mantle and Willy Mays were there, would it make any difference who you chose?

[studiot]

2 hours ago, guidoLamoto said:

We agree about Newton's place in history, but it's difficult to fault Leonardo- scientist, artist and futurist.

If we were chosng up sides for a stick ball game in the streets and Mickey Mantle and Willy Mays were there, would it make any difference who you chose?

Well as I have no idea who or what they are I really don't have a choice.

As to leonardo, what Maths is he responsible for ?, apart from a few extensions to Euclid ?

[guidoLamoto]

You don't know Mantle or Mays?... I take it you're either not American or some kind of Communist or Vegetarian.. (They were excellent baseball players, both palying in New York in the '50s. Add in Snider playing for Brooklyn in that decade and you have the meat for endless hours of arguments among sports fans in the bars over who was the best center fielder.)

Leonardo had as much to do with math as Sir Isaac had to do with sculpturing and painting...although Leonardo's plan for a bridge over the strait at Constantinople has been shown in models  to be feasable and mechanically sound engineering, far advanced for its time. https://www.livescience.com/da-vinci-bridge-never-made.html

He was also said to be the strongest man in Italy. .. and I think he invented the Whoopie Cushion too. A well rounded man. Newton can't compete on that level. 

[StringJunky]

30 minutes ago, studiot said:

Well as I have no idea who or what they are I really don't have a choice.

As to leonardo, what Maths is he responsible for ?, apart from a few extensions to Euclid ?

... apart from a few extensions to Euclid? And how many extensions to the mathematical corpus have you done?  

 

12 minutes ago, guidoLamoto said:

You don't know Mantle or Mays?... I take it you're either not American or some kind of Communist or Vegetarian.. (They were excellent baseball players, both palying in New York in the '50s. Add in Snider playing for Brooklyn in that decade and you have the meat for endless hours of arguments among sports fans in the bars over who was the best center fielder.)

Leonardo had as much to do with math as Sir Isaac had to do with sculpturing and painting...although Leonardo's plan for a bridge over the strait at Constantinople has been shown in models  to be feasable and mechanically sound engineering, far advanced for its time. https://www.livescience.com/da-vinci-bridge-never-made.html

He was also said to be the strongest man in Italy. .. and I think he invented the Whoopie Cushion too. A well rounded man. Newton can't compete on that level. 

If it was a choice between being stuck on an island with either, Leonardo would be my preferred survival partner.

[joigus]

"Differentiation" is frequently used as synonymous of "taking the derivative". Interpreted in that sense, it is an operation which, given a function of the right class (differentiable functions) produces its derivative, which is another function. Example:

 

\[ x^{3}\overset{\frac{d}{dx}}{\mapsto}3x^{2} \]

We say that we have "differentiated" \( x^{3} \) to obtain \( 3x^{2} \), its "derivative". 3xerivative".

Mathematicians sometimes talk about a function being "differentiable" when you can express little increments of it as a linear function of the increment in its variable. This linear function of the increment is what they (the mathematicians) call "the differential". With our previous example, the difference between the values of \( x^{3} \) evaluated at \( x+h \) and the same function evaluated at \( x \) is,

\[ x^{3}+3x^{2}h+3xh^{2}+h^{3}-x^{3}=\left(3x^{2}\right)h+\left(3x+h\right)h^{2} \]

Now, the idea is that, when \( h \), the increment in the independent variable, is very small, the increment in the dependent function is a linear function plus something "very small". And indeed,

\[ \left(x+h\right)^{3}-x^{3}=\left(3x^{2}\right)h+o\left(h^{2}\right) \]

where, \( o\left(h^{2}\right)=\left(3x+h\right)h^{2} \) really is "something very small" when \( h \) itself is small. This is what physicists write as,

That's why physicists like to write,

\[ y=x^{3} \]

\[ dy=\frac{dy}{dx}dx=3x^{2}dx \]

when rigorous mathematicians would rather have us write something like,

\[ \triangle y=3x^{2}\triangle x+o\left(\left(\triangle x\right)^{2}\right) \]

Now, a function has a derivative at \( x \) if and only if there increments of it can be expressed as a linear function of the increment plus a little correction that goes to zero as the increment goes to zero.

Did that help at all? If not, please just ignore it.

[Zakher]

Alright, let's imagine you're playing a game where you have to count how many steps you take. Every time you take a step, you mark it on a counter.

So, "differentiation" is like figuring out how fast you're taking those steps. For example, if you take 5 steps in 1 minute, your speed is 5 steps per minute.

Now, "differential" is how much you changed your speed at a specific moment. For instance, if you start walking faster or slower, that's a change in your differential.

And you know what's cool? Understanding this stuff once really helped me out. It's like having a superpower to analyze and make sense of how things are changing. So, differentiation is about determining the pace of your actions, and the differential is the change in that pace at a particular moment. It might sound a bit tricky at first, but once it clicks, it's pretty awesome!