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MathGroup Archive 2007

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Re: differentiate a function of a function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75110] Re: differentiate a function of a function
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Mon, 16 Apr 2007 20:27:51 -0400 (EDT)
  • References: <evvbpf$bc1$1@smc.vnet.net>

> 1) to say that f is a function: f(x,y) = Tan[t(x,y)+a(x,y)]

Hi I assume you know to define simple functions in Mathematica.
So something like the following does not need further explanation.

In[1]:=
f[x_, y_] := Tan[t[x, y] + a[x, y]]

> 2) take D[f,x] , such that also t(x,y) and a(x,y) will be also
> differentiated

Here are some examples:

In[5]:=
D[f[x, y], x]
Integrate[%, x] == f[x, y]

Out[5]=
Sec[a[x, y] + t[x, y]]^2*(Derivative[1, 0][a][x, y] + Derivative[1, 0]
[t][x, y])
Out[6]=
True

In[10]:=
D[f[x, y], {x, 2}]
Integrate[%, x, x] == f[x, y]

Out[10]=
2*Sec[a[x, y] + t[x, y]]^2*Tan[a[x, y] + t[x, y]]*(Derivative[1, 0][a]
[x, y] + Derivative[1, 0][t][x, y])^2 +
  Sec[a[x, y] + t[x, y]]^2*(Derivative[2, 0][a][x, y] + Derivative[2,
0][t][x, y])
Out[11]=
True

In[13]:=
D[f[x, y], x, y]
Integrate[%, x, y] == f[x, y]

Out[13]=
2*Sec[a[x, y] + t[x, y]]^2*Tan[a[x, y] + t[x, y]]*(Derivative[0, 1][a]
[x, y] + Derivative[0, 1][t][x, y])*
   (Derivative[1, 0][a][x, y] + Derivative[1, 0][t][x, y]) + Sec[a[x,
y] + t[x, y]]^2*
   (Derivative[1, 1][a][x, y] + Derivative[1, 1][t][x, y])
Out[14]=
True

In[18]:=
D[f[x, y], {y, 3}]
Integrate[%, y, y, y] == f[x, y]

Out[18]=
2*Sec[a[x, y] + t[x, y]]^4*(Derivative[0, 1][a][x, y] + Derivative[0,
1][t][x, y])^3 +
  4*Sec[a[x, y] + t[x, y]]^2*Tan[a[x, y] + t[x, y]]^2*(Derivative[0, 1]
[a][x, y] + Derivative[0, 1][t][x, y])^3 +
  6*Sec[a[x, y] + t[x, y]]^2*Tan[a[x, y] + t[x, y]]*(Derivative[0, 1]
[a][x, y] + Derivative[0, 1][t][x, y])*
   (Derivative[0, 2][a][x, y] + Derivative[0, 2][t][x, y]) + Sec[a[x,
y] + t[x, y]]^2*
   (Derivative[0, 3][a][x, y] + Derivative[0, 3][t][x, y])
Out[18]=
True

and so on...

> 3) be able to substitute these into some equation like: f_x f_xy = 8

No clear what do you mean.

Regards
Dimitris

=CF/=C7 kem =DD=E3=F1=E1=F8=E5:
> Hi,
>
> I was wondering how do I define a function in mathematica to be able
> to differentiate it etc, where some of the parameters of this function
> should be also a function. For example I want to be able to do the
> following operations:
>
> 1) to say that f is a function: f(x,y) = Tan[t(x,y)+a(x,y)]
>
> 2) take D[f,x] , such that also t(x,y) and a(x,y) will be also
> differentiated
>
> 3) be able to substitute these into some equation like: f_x f_xy = 8
>
> Thanks a lot
>
> kem



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