What is the difference between $ dy $ and $ Delta y $?
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I'm learning to approximate functions using derivatives. I'm a little confused about the notation(s) $ dx $ (or $ dy $) and $ Delta x $ (or $ Delta y $). Do they mean the exact same thing -- except for the different curves? In other words, does $ dx $ represent the (small) increase in $ x $ for the straight line given in the image and $ Delta x $ represent the (small) incrase in $ x $ for the given curve?
Do we use $ dx $ & $ Delta x $, and $ dy $ & $Delta y$ just to differentiate between the increments in the two different function or are they different?
Why is $ dx $ called the "differential" of $ x $ and $ Delta x $ being called the "increment" of $ x $ (similarly for y)? My textbook also states:
We may note that the differential of the dependent variable is not
equal to the increment of the variable where as the differential of
independent variable is equal to the increment of the variable
calculus derivatives
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
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|
show 1 more comment
$begingroup$
I'm learning to approximate functions using derivatives. I'm a little confused about the notation(s) $ dx $ (or $ dy $) and $ Delta x $ (or $ Delta y $). Do they mean the exact same thing -- except for the different curves? In other words, does $ dx $ represent the (small) increase in $ x $ for the straight line given in the image and $ Delta x $ represent the (small) incrase in $ x $ for the given curve?
Do we use $ dx $ & $ Delta x $, and $ dy $ & $Delta y$ just to differentiate between the increments in the two different function or are they different?
Why is $ dx $ called the "differential" of $ x $ and $ Delta x $ being called the "increment" of $ x $ (similarly for y)? My textbook also states:
We may note that the differential of the dependent variable is not
equal to the increment of the variable where as the differential of
independent variable is equal to the increment of the variable
calculus derivatives
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
$endgroup$
1
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
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– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
1
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If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
1
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08
|
show 1 more comment
$begingroup$
I'm learning to approximate functions using derivatives. I'm a little confused about the notation(s) $ dx $ (or $ dy $) and $ Delta x $ (or $ Delta y $). Do they mean the exact same thing -- except for the different curves? In other words, does $ dx $ represent the (small) increase in $ x $ for the straight line given in the image and $ Delta x $ represent the (small) incrase in $ x $ for the given curve?
Do we use $ dx $ & $ Delta x $, and $ dy $ & $Delta y$ just to differentiate between the increments in the two different function or are they different?
Why is $ dx $ called the "differential" of $ x $ and $ Delta x $ being called the "increment" of $ x $ (similarly for y)? My textbook also states:
We may note that the differential of the dependent variable is not
equal to the increment of the variable where as the differential of
independent variable is equal to the increment of the variable
calculus derivatives
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
$endgroup$
I'm learning to approximate functions using derivatives. I'm a little confused about the notation(s) $ dx $ (or $ dy $) and $ Delta x $ (or $ Delta y $). Do they mean the exact same thing -- except for the different curves? In other words, does $ dx $ represent the (small) increase in $ x $ for the straight line given in the image and $ Delta x $ represent the (small) incrase in $ x $ for the given curve?
Do we use $ dx $ & $ Delta x $, and $ dy $ & $Delta y$ just to differentiate between the increments in the two different function or are they different?
Why is $ dx $ called the "differential" of $ x $ and $ Delta x $ being called the "increment" of $ x $ (similarly for y)? My textbook also states:
We may note that the differential of the dependent variable is not
equal to the increment of the variable where as the differential of
independent variable is equal to the increment of the variable
calculus derivatives
calculus derivatives
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
asked Dec 30 '18 at 11:21
WorldGovWorldGov
345212
345212
1
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
1
$begingroup$
If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
1
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08
|
show 1 more comment
1
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
1
$begingroup$
If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
1
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08
1
1
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
1
1
$begingroup$
If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
$begingroup$
If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
1
1
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08
|
show 1 more comment
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1
$begingroup$
An increment is a finitely small change while a differential is an infinitely small change.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
$begingroup$
Never read theory from NCERT.
$endgroup$
– harshit54
Dec 30 '18 at 11:56
1
$begingroup$
If a finite change $Delta x$ in $x$ causes a change $Delta y$ in $y$, then $frac{dy}{dx}$ is defined as $lim_{Delta yrightarrow 0}frac{Delta y}{Delta x}$ —i.e. $Delta y$ is finite but $dy$ isn't.
$endgroup$
– timtfj
Dec 30 '18 at 17:33
1
$begingroup$
Putting it another way, $dx$ is (roughly speaking) an infinitely small $Delta x$.
$endgroup$
– timtfj
Dec 30 '18 at 17:39
$begingroup$
Sorry, that should have been $frac{dy}{dx}=lim_{Delta xrightarrow 0}frac{Delta y}{Delta x}$
$endgroup$
– timtfj
Dec 30 '18 at 18:08