Anonymous at Fri, 1 Nov 2024 16:26:08 UTC No. 16458355

>>16458275
You define partial derivatives first, then you use them as part of the definition of the differential. The partial derivatives is basic chain rule 1-D calculus, proven in your first calc class using limits. Differentials come after partials, not the other way around

Anonymous at Fri, 1 Nov 2024 16:43:52 UTC No. 16458375

>>16458275
use definition of derivative to prove it. this is bullshit. also you should assume f and g are continuous at all points x element of R0

Anonymous at Fri, 1 Nov 2024 16:51:12 UTC No. 16458387

>>16458275
“Total derivative” you’re assuming your conclusion OP. Use the definition of derivative for single variable functions first

Anonymous at Fri, 1 Nov 2024 22:47:31 UTC No. 16458973

>>16458355
>>16458387
but do you really need the peoduct rule to define total and partial derivative?
>>16458375
yeah i know it's bullshit I need to use limit definition right?

Anonymous at Sat, 2 Nov 2024 06:35:44 UTC No. 16459354

>>16458275
rigor isnt defined in mathematics

Anonymous at Sat, 2 Nov 2024 07:53:35 UTC No. 16459416

>>16458275
I recommend using Einstein notation when hand writing. Makes life a ton easier.

Anonymous at Sat, 2 Nov 2024 12:06:48 UTC No. 16459561

>>16459416
>when hand writing
what other ways to write are there?

Anonymous at Sun, 3 Nov 2024 13:31:41 UTC No. 16460871

>>16458275
>dh = dh/dx * dx + dh/dy * dy
>dh = dh*1 + dh*1
>1 = 2
undefined bros...

Anonymous at Sun, 3 Nov 2024 14:03:24 UTC No. 16460897

>>16460871
Come back when you've done multivariate calculus.

The curly dx's (partials) are not the same as the normal dx that you see in single variable calculus.

Anonymous at Sun, 3 Nov 2024 16:02:25 UTC No. 16461045

>>16460897
I'm so sorry your handwriting is so shit I couldn't even see the difference between the dels and dees

Anonymous at Mon, 4 Nov 2024 15:15:03 UTC No. 16462193

>>16460871
>>16461045
no you are both right
my handwriting is shit because it isnt my handwriting, it is literally drawn in paint

also, I dropped the partial delta symbol on line 2 after merely applying the definition because the functions are only dependent on one variable so the partial is the same as the normal derivative

as for why the fractions do not cancel out, that is because they are not truly factions: leibniz notation is 100% notational abuse but it just werks (if u know what u are doing)
as an example, if u obttain dh = 2 * dh you have abused it too much.
but it is perfectly fine to write
dh = g df + f dg