Here are some ways.

Consider 

z:=x*diff(y(x),x)^3;
                                     3
                           / d      \ 
                         x |--- y(x)| 
                           \ dx     / 

  Suppose we want the derivative with respect to diff(y(x),x).

  Maybe is the easiest way is to use the diff command in the Physics package

Physics:-diff(z,diff(y(x),x));
                                      2
                            / d      \ 
                        3 x |--- y(x)| 
                            \ dx     / 

  Another way is to use frontend

frontend(diff, [z,diff(y(x),x)]);
                                      2
                            / d      \ 
                        3 x |--- y(x)| 
                            \ dx     / 

  And another way is to use subs

subs(diff(y(x),x)=m,z);
                                 3
                              x m 
diff(%,m);
                                  2
                             3 x m 
subs(m=diff(y(x),x),%);
                                      2
                            / d      \ 
                        3 x |--- y(x)| 
                            \ dx     / 

According the the help pages

http://www.maplesoft.com/support/help/Maple/view.aspx?path=collect

"The indeterminates can be names or unevaluated function calls but not sums or products or fractional powers"

I don't notice any difference than usual from here (Leeds, UK). It's not the fastest site on the net, but I also don't find it particularly slow either. Could it be internet congestion in your area ?

You have to be careful about the point at which expressions are evaluated..

Does this give you the result you are expecting ? 

restart:

x := .5:
T := 150:
nA := 100:
TA := 1:
CA := 100:
TB := 30:
CB := 1:
T1 := 1:
C1 := .1:
T3 := 600:
C3 := 10:
T4 := 0.1e7:
C4 := 1:
T5 := 200:
C5 := 10:
T6 := 300:
C6 := 50:
krWL := 10:
krQD := 1:
a := CA*exp(-TA/T): 
b := CB*exp(-TB/T): 
c := C1*exp(-T1/T): 
d := C3*exp(-T3/T): 
e := C4*exp(-T4/T): 
f := krWL: 
g := krQD: 
h := nA:
k := C5*exp(-T5/T):
l := C6*exp(-T6/T): 
G := 10^x:


sys:=0 = -a*(h-Y-X-Z-V+W)+b*Z, 0 = k*Y-l*V, 

0 = d*Y-e*X-g*W*X/(W+X), 0 = a*(h-Y-X-Z-V+W)+G-c*Z-b*Z, 0 = c*Z-d*Y+e*X-k*Y+l*V-f*W*Y/(W+Y):

solve({sys},{V, W, X, Y, Z});
{V = 0.7088373557, W = 0.3349788314, X = 65.70981532, Y = 1.819646165, 

                     /                  -2894                     -2894  
  Z = 31.83429940}, { V = 1.129437854 10     , W = -2.899363646 10     , 
                     \                                                   

                                     -2894                 \    
  X = 67.90332001, Y = 2.899363646 10     , Z = 31.83429940 }, {
                                                           /    

  V = 18.73254561, W = -0.5579850222, X = 0.5247413775, Y = 48.08804800, 

  Z = 31.83429940}, {V = 0.1204251010, W = -67.41717963, X = 0.05657373428, 

  Y = 0.3091415421, Z = 31.83429940}

 You may find it better to make a set of parameters and use eval before calling solve

Does convert/phaseamp do what you want ? Based on the first part of your code:

aa:= 0.0000000000000000277555756156289135105908+VERTOFFmain_1+VERTOFFaad_1+((AMPmain_1+AMPaaa_1)*0.4330127018922191872718485683435574173927)*cos((FREQmain_1+FREQaab_1)* 1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) +((AMPmain_1+AMPaaa_1)*0.4330127018922194648276047246326925233006)*sin((FREQmain_1+FREQaab_1)*1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) +((AMPmain_1+AMPaaa_1));

       aa :=      1.+.4330127018922191872718485683435574173927*cos(1.0*t)+.4330127018922194648276047246326925233006*sin(1.0*t)

convert(aa,phaseamp,t);

     1.+.6123724357*cos(1.0*t-.7853981634)

I think your problem is that yc is not smooth at x=0, x=0.00576 and x=0.12, hence the derivative does not exist at those points.

Also, you should either use arctan() or MTN[atan]() for the inverse tangent function

You are asking Maple to evaluate an integral but you are not telling it which variable to integrate with respect to.

Inert integation will do what you are asking, though I don't know if it is really what you want. If you are integrating wrt x:

D1 := -(1/4)*(Int(f*exp(-v*s)*(1+v*s)/v^3, x = 0 .. infinity));

Do you really mean diff(x(t),[x$2]) at the start of EOM ? Shouldn't it be diff(x(t),t$2) ?

Also, there looks to be some occurances of multiplication without the *, which may be causing problems. I find it's always best to use * and not rely on implicit multiplication.

Can you give a bit more detail about your data ? Since it's a time series, why are you using polynomial interpolation ? Is there a physical reason ? Are the data seasonal ? You might want to try statistical methods for time smoothing, modeling the seasonal factors, autocorellations and heteroskedasticity.

I don't know how to return only a fixed number of solutions, but if you use fsolve, you can pass an interval to restrict the range of solutions it returns....

http://www.maplesoft.com/support/help/Maple/view.aspx?path=fsolve/details

What kind of data is it ? Maybe you could model it, and take the derivative of the model ?

I expect you will have to write your own function to do that.....

I guess it will be tricky to get coefficients that are zero, but other than that, I thought dcoeffs does what you want ?

with(PDEtools):
declare():
pde:=2*diff(U(x,y),x$2)+6*diff(diff(U(x,y),x),y)+3*diff(U(x,y),x)+2*U(x,y);
dcoeffs(pde,U(x,y));
                                   6, 2, 3, 2 

You could use the Mean function in the statistics package

Statistics[Mean]([4,6,8]);

6

For more info

http://www.maplesoft.com/support/help/Maple/view.aspx?path=Statistics/Mean

I like

P. Billingsley  "Probability and Measure"

http://www.amazon.com/Probability-Measure-3rd-Patrick-Billingsley/dp/0471007102/ref=pd_sim_b_2

It is rigorous.