Re: differentiate a function of a function

• To: mathgroup at smc.vnet.net
• Subject: [mg75080] Re: differentiate a function of a function
• From: "Norbert Marxer" <marxer at mec.li>
• Date: Mon, 16 Apr 2007 20:12:30 -0400 (EDT)
• References: <evvbpf\$bc1\$1@smc.vnet.net>

```On 16 Apr., 10:23, "kem" <keme... at gmail.com> wrote:
> 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

Hello

If you evaluate the following:

f[x_, y_] = Tan[a[x, y] + t[x, y]]
D[f[x, y], x]
D[f[x, y], x] + D[f[x, y], x, y] == 8

you will get:

Out[1]=
Tan[a[x, y] + t[x, y]]

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

Out[3]=
Sec[a[x, y] + t[x, y]]^2*(Derivative[1, 0][a][x, y] +
Derivative[1, 0][t][x, y]) +
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]) == 8

If you run the select the above cells (in Mathematica) and convert