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