In[]:=
K[s_,alpha_,beta1_,beta2_]:=-alpha*(Log[1-s/beta1]+Log[1-s/beta2])dKds[s_,alpha_,beta1_,beta2_]=D[K[s,alpha,beta1,beta2],s]Simplify[dKds[s,alpha,beta1,beta2]]Solve[dKds[s,alpha,beta1,beta2]==x,s,Reals]
Out[]=
-alpha--
1
beta11-
s
beta1
1
beta21-
s
beta2
Out[]=
-
alpha(beta1+beta2-2s)
(beta1-s)(-beta2+s)
Out[]=
s-,s+
-2alpha+beta1x+beta2x
2x
1
2
4+-2beta1beta2+
2
alpha
2
beta1
2
x
2
x
2
beta2
2
x
2
x
-2alpha+beta1x+beta2x
2x
1
2
4+-2beta1beta2+
2
alpha
2
beta1
2
x
2
x
2
beta2
2
x
2
x
In[]:=
D[K[s,alpha,beta1,beta2],{s,2}]
Out[]=
-alpha--
1
2
beta1
2
1-
s
beta1
1
2
beta2
2
1-
s
beta2