On 13 Feb 2007 07:00:02 -0000, Scott Ballantyne <sdb at ssr.com> wrote:
>
> Experimenting with maxima for the first time, I tried to compute the
> derivative of (%e^x-%e^(-x))/(%e^x+%e^(-x)), which I believe should
> be: 4/(e^x+e^-x)^2.
>
> Maxima gives 1 - (e^x-e^(-x))^2/(e^x+e^(-x))^2
>
> which is of course fine, but when I try to simplify this (I used
> ratsimp(%) and then factor(%)), maxima delivers
>
> 4e^(2x)/(e^(2x)+1)^2
>
> Perhaps this is correct...
>
As previous writers have mentioned, there are *many* correct forms. Here
are a few, in increasing order of length. Pick your favorite.
sech(x)^2
1/cosh(x)^2
1-tanh(x)^2
2/(cosh(2*x)+1)
{4/(%e^x+%e^-x)^2
1-sinh(x)^2/cosh(x)^2
-(tanh(x)-1)*(tanh(x)+1)
4*%e^(2*x)/(%e^(2*x)+1)^2
1-(%e^x-%e^-x)^2/(%e^x+%e^-x)^2
-(sinh(x)^2-cosh(x)^2)/cosh(x)^2
4*%e^(2*x)/(%e^(4*x)+2*%e^(2*x)+1)
-(sinh(x)-cosh(x))*(sinh(x)+cosh(x))/cosh(x)^2
-sech(x)^2*(-cosh(2*x)/2-(2-2*cosh(2*x))/4-1/2)
4*(sinh(2*x)+cosh(2*x))/(sinh(2*x)+cosh(2*x)+1)^2
-4*((%e^x-%e^-x)^2/4-(%e^x+%e^-x)^2/4)/(%e^x+%e^-x)^2
4*(sinh(2*x)+cosh(2*x))/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)
4*(sinh(2*x)+cosh(2*x))/(sinh(4*x)+cosh(4*x)+2*(sinh(2*x)+cosh(2*x))+1)
(4*sinh(2*x)+4*cosh(2*x))/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)
-4*((%e^x-%e^-x)/2-(%e^x+%e^-x)/2)*((%e^x+%e^-x)/2+(%e^x-%e^-x)/2)/(%e^x+%e^-x)^2
-1/(%e^(4*x)+2*%e^(2*x)+1)-%e^(2*x)/(%e^(2*x)+%e^-(2*x)+2)+2/(%e^(2*x)+%e^-(2*x)+2)+1
4*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)/(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2+1)^2
(4*sinh(2*x)+4*cosh(2*x))/(sinh(2*x)^2+(2*cosh(2*x)+2)*sinh(2*x)+cosh(2*x)^2+2*cosh(2*x)+1)
-%e^(4*x)/(%e^(4*x)+2*%e^(2*x)+1)+2*%e^(2*x)/(%e^(4*x)+2*%e^(2*x)+1)-1/(%e^(4*x)+2*%e^(2*x)+1)+1
4*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)/((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2+1)^2
-1/(sinh(4*x)+cosh(4*x)+2*(sinh(2*x)+cosh(2*x))+1)-(sinh(2*x)+cosh(2*x))/(2*cosh(2*x)+2)+2/(2*cosh(2*x)+2)+1
4*sinh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+4*cosh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)
4*(sinh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+cosh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1))
-1/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-sinh(2*x)/(2*cosh(2*x)+2)-cosh(2*x)/(2*cosh(2*x)+2)+1/(cosh(2*x)+1)+1
-1/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-sinh(2*x)/(2*cosh(2*x)+2)-cosh(2*x)/(2*cosh(2*x)+2)+2/(2*cosh(2*x)+2)+1
4*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)/((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)+1)
4*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)/(sinh(x)^4+4*cosh(x)*sinh(x)^3+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4+1)
4*sinh(2*x)/(sinh(2*x)^2+2*cosh(2*x)*sinh(2*x)+2*sinh(2*x)+cosh(2*x)^2+2*cosh(2*x)+1)+4*cosh(2*x)/(sinh(2*x)^2+2*cosh(2*x)*sinh(2*x)+2*sinh(2*x)+cosh(2*x)^2+2*cosh(2*x)+1)
-((sinh(2*x)-cosh(2*x)-4)*sinh(4*x)+(sinh(2*x)-cosh(2*x)-4)*cosh(4*x)+2*sinh(2*x)^2-7*sinh(2*x)-cosh(2*x)*(2*cosh(2*x)+7)-2)/(2*(cosh(2*x)+1)*(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1))
-(sinh(4*x)+cosh(4*x))/(sinh(4*x)+cosh(4*x)+2*(sinh(2*x)+cosh(2*x))+1)+2*(sinh(2*x)+cosh(2*x))/(sinh(4*x)+cosh(4*x)+2*(sinh(2*x)+cosh(2*x))+1)-1/(sinh(4*x)+cosh(4*x)+2*(sinh(2*x)+cosh(2*x))+1)+1
-1/((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)+1)-((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)/(%e^(2*x)+%e^-(2*x)+2)+2/(%e^(2*x)+%e^-(2*x)+2)+1
-((sinh(2*x)-cosh(2*x)-4)*sinh(4*x)+(sinh(2*x)-cosh(2*x)-4)*cosh(4*x)-7*sinh(2*x)-7*cosh(2*x)-4)/((2*cosh(2*x)+2)*sinh(4*x)+(2*cosh(2*x)+2)*cosh(4*x)+(4*cosh(2*x)+4)*sinh(2*x)+4*cosh(2*x)^2+6*cosh(2*x)+2)
-(sinh(2*x)*sinh(4*x)-cosh(2*x)*sinh(4*x)-4*sinh(4*x)+sinh(2*x)*cosh(4*x)-cosh(2*x)*cosh(4*x)-4*cosh(4*x)+2*sinh(2*x)^2-7*sinh(2*x)-2*cosh(2*x)^2-7*cosh(2*x)-2)/(2*(cosh(2*x)+1)*(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1))
-1/(sinh(x)^4+4*cosh(x)*sinh(x)^3+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4+1)-(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)/(2*(sinh(x)^2+cosh(x)^2)+2)+2/(2*(sinh(x)^2+cosh(x)^2)+2)+1
-((sinh(2*x)-cosh(2*x)-4)*sinh(4*x)+(sinh(2*x)-cosh(2*x)-4)*cosh(4*x)+2*sinh(2*x)^2-7*sinh(2*x)-2*cosh(2*x)^2-7*cosh(2*x)-2)/((2*cosh(2*x)+2)*sinh(4*x)+(2*cosh(2*x)+2)*cosh(4*x)+(4*cosh(2*x)+4)*sinh(2*x)+4*cosh(2*x)^2+6*cosh(2*x)+2)
-sinh(4*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-cosh(4*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+2*sinh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+2*cosh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-1/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+1
2*(sinh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+cosh(2*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1))-sinh(4*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-cosh(4*x)/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)-1/(sinh(4*x)+cosh(4*x)+2*sinh(2*x)+2*cosh(2*x)+1)+1}
-((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2)/((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)+1)+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)/((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)+1)-1/((%e^(4*x)+%e^-(4*x))/2+(%e^(4*x)-%e^-(4*x))/2+2*((%e^(2*x)+%e^-(2*x))/2+(%e^(2*x)-%e^-(2*x))/2)+1)+1
-(sinh(x)^4+4*cosh(x)*sinh(x)^3+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4)/(sinh(x)^4+4*cosh(x)*sinh(x)^3+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4+1)+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)/(sinh(x)^4+4*cosh(x)*sinh(x)^3+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4+1)-1/(sinh(x)^4+4*cosh(x)*sinh(x)^3+2*(sinh(x)^2+2*cosh(x)*sinh(x)+cosh(x)^2)+6*cosh(x)^2*sinh(x)^2+4*cosh(x)^3*sinh(x)+cosh(x)^4+1)+1
These were generated by:
ops:'[factor,trigify,radcan,rectform,polarform,trigsimp,trigrat,trigreduce,trigexpand,factorsum,ratsimp,exponentialize]$
moreforms(forms) :=
xreduce('union,makelist(setify(makelist((q)(r),q,ops)),r,listify(forms)))$
forms:{1-(%e^x-1/%e^x)^2/(%e^x+1/%e^x)^2};
forms: moreforms(forms);
(iterate as much as you want...)
Have fun
-s