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Re: Further expanding d[xy]/dx in the output of a Dt calculation

• To: mathgroup at smc.vnet.net
• Subject: [mg112996] Re: Further expanding d[xy]/dx in the output of a Dt calculation
• From: telefunkenvf14 <rgorka at gmail.com>
• Date: Sun, 10 Oct 2010 06:41:53 -0400 (EDT)
• References: <i8pghf$fit$1@smc.vnet.net>

On Oct 9, 5:35 am, John Accardi <acca... at accardi.com> wrote:
>   Hello ... any help appreciated ....
>
> Ref:
>
> http://www.accardi.com/ImplicitDifferentiationForumQuery.nb
>
> As you see in the *.nb cited above, during the implicit differentiation
> calculation, Mathematica includes d[xy]/dx in the solution.  I would
> like the product differentiation rule invoked for a complete
> differentiation during the Dt calculation.  Then, I would hope to see
> d[xy]/dx not show in the value of dy/dx in the Solve output.
>
> Here it is in ASCII, but it might be easier to see if you run the
> notebook ....
>
> eq5 = x^3-x^2y+2xy^2 == 12
> Dt[eq5,x]
> -3x^2 dy/dx +3x^2 + 4xy dxy/dx -6xy = 0
> Solve[%%,dy/dx]
> {{dy/dx -> (3x^2 + 4xy dxy/dx -6xy)/3x^2}}
>
> How can I get Mathematica to show a more complete differentiation by
> expanding dxy/dx?
>
> Thank you
> Gianni

Mathematica is treating 'xy' as a unique variable, not a product.
Include a space between the variables OR use * to indicate
multiplication in your 'eq5' and I think you'll get what you want.

FYI, I looked at your notebook, and noticed you were using traditional
form input and output. Everyone has their own prefs here, but my
experience is that using StandardForm input/output helps you catch
your mistakes, making programming much less painful!! (You can always
make things pretty later...)

In[1]:= eq5 = x^3 - 3 x^2 y + 2 x y^2 == 12
Dt[eq5, x]
Solve[%, Dt[y, x]]

Out[1]= x^3 - 3 x^2 y + 2 x y^2 == 12
Out[2]= 3 x^2 - 6 x y + 2 y^2 - 3 x^2 Dt[y, x] + 4 x y Dt[y, x] == 0
Out[3]= {{Dt[y, x] -> (3 x^2 - 6 x y + 2 y^2)/(x (3 x - 4 y))}}

-RG