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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:45:50
What if u is like an entire equation
1
0
544,714,890,858,856,450
iLuvPlayboiCarti
7264
iLuvPlayboiCarti
false
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:45:55
Like 2x+ 7
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:45:57
?
0
1
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:00
It can be an entire expression
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:46:03
Is that still u
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:05
thats fine
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:09
you can set u to be anything
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:46:10
I meant exoression
1
0
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:21
as long as you know how to differentiate after you substitute
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:25
thats why the chain rule is so useful
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:26
$$\dv{y}{x} = \dv{x}\left(u^{-1}\right)$$
0
1
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:46:29
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:30
So now we have this
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:41
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:45
Now, I apply the chain rule
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:46:58
$$\dv{y}{x} = \dv{u}{u}\dv{x}\left(u^{-1}\right)$$
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:47:06
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:47:11
I usually show this step first (multiplying by 1)
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:47:13
makes sense?
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:47:33
Then the chain rule says we can treat these like fractions. I will swap the 'du' and 'dx' on the 'bottom' of the 'fraction'
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:47:35
Hmmm
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:47:41
They're not actually fractions, but the chain rule says we can do this.
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:47:53
$$\dv{y}{x} = \dv{u}{x}\dv{u}\left(u^{-1}\right)$$
0
1
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:47:55
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:10
You can skip this step if you want
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:15
But I find it usually helps myself
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:48:15
Yea
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:25
Now we can just differentiate the right thing
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:36
$$\dv{y}{x} = \dv{u}{x}\left(-u^{-2}\right)$$
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:48:38
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:40
ok?
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:48:47
Yea
1
0
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:47
So that's the first thing out of the way
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:53
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:48:56
And I substitute u back in
0
1
null
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:48:58
Can u do it without substituting it to u
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:49:06
?
1
0
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:10
Yes I can
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:10
But it's not so obvious
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:24
I can show if you want
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:42
But I think it'll just end up confusing if I show without substituting
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:49:42
Yea il cope
1
0
null
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:59
We had this
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:49:59
$$\dv{y}{x} = \dv{x}\left((e^{x^2})^{-1}\right)$$
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:50:01
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:19
$$\dv{y}{x} = \dv{\left(e^{x^2}\right)}{\left(e^{x^2}\right)}\dv{x}\left((e^{x^2})^{-1}\right)$$
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:50:22
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:28
Now I essentially do the same thing
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:31
with what u was
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:42
$$\dv{y}{x} = \dv{\left(e^{x^2}\right)}{x}\dv{\left(e^{x^2}\right)}\left((e^{x^2})^{-1}\right)$$
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:50:45
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:53
Like this is kinda hard to work with ngl
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:50:58
I'd prefer to just sub u
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:01
and make things clear
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:51:04
Wait
1
0
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:51:17
https://cdn.discordapp.c…20123_005115.jpg
20220123_005115.jpg
1
0
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:22
?
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:51:28
Wouldn't u be x^2
1
0
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:38
what you mean in the next step?
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:41
yes
0
1
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:53
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:51:58
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:01
So I had this
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:04
and I will sub u back in
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:52:12
So isn't it just e^x^2 times 2x
1
0
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:22
sure, but I really advise against remembering formulas
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:26
and just remember how to do the chain rule
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:31
once you get practice its pretty quick
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:41
(you don't actually write the substitution down and just do it in your head)
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:52:52
$$\dv{y}{x} = \dv{\left(e^{x^2}\right)}{x}\left(-\left(e^{x^2}\right)^{-2}\right)$$
0
1
null
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null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:53:06
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:53:09
Because the formula you showed just comes from the chain rule
0
1
null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:53:22
Okay so what exactly happenes to the negative 1 in the exponent?
1
0
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:53:32
It dropped because of the
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:53:38
Chain rule
1
0
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:53:39
?
1
0
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:53:42
$$nx^{n-1}$$
0
1
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:53:47
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:53:51
This?
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:54:02
Isn't that chain rule
1
0
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:11
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:19
No, this is exponent rule
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:23
This one is
0
1
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:54:25
Oh yea nvm
1
0
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:32
And so here it droped to -2
0
1
null
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null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:35
and -1 got multiplied
0
1
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null
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:54:44
So the exponent changes to -2
1
0
null
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:44
Which leaves me with this
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:44
$$\dv{y}{x} = \dv{\left(e^{x^2}\right)}{x}\left(-\left(e^{x^2}\right)^{-2}\right)$$
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:54:47
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:48
Yes
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
401,842,339,590,570,000
PRIT
01/23/2022 00:54:54
Okay
1
0
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:54:56
I will just clean this up first
0
1
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:55:22
$$\dv{y}{x} = -\dv{\left(e^{x^2}\right)}{x}e^{-2x^2}$$
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:55:25
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:55:33
x^2 and -2 get multiplied
0
1
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null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:55:35
simplifying, ok?
0
1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:55:50
Next I set $u = x^2$
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1
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205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
510,789,298,321,096,700
TeXit
01/23/2022 00:55:52
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
null
null
205,594
help-1
f2ff3d77869e4ffbba492e650c969a8c
321,445,119,171,756,000
Shuri2060
01/23/2022 00:56:02
$$\dv{y}{x} = -\dv{\left(e^{u}\right)}{x}e^{-2x^2}$$
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1
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