 | (a)
In[1]:=
y = E^ArcSin[x]
D[y, {x, 1}]
Out[1]=
E^ArcSin[x]
Out[2]=
E^ArcSin[x]/Sqrt[1 - x^2]
In[3]:=
x*D[y, {x, 1}]
Out[3]=
(E^ArcSin[x]*x)/Sqrt[1 - x^2]
In[4]:=
D[y, {x, 2}]
Out[4]=
(E^ArcSin[x]*x)/(1 - x^2)^(3/2) +
E^ArcSin[x]/(1 - x^2)
In[5]:=
Simplify[(1 - x^2)*D[y, {x, 2}]]
Out[5]=
E^ArcSin[x]*(1 + x/Sqrt[1 - x^2])
In[6]:=
Simplify[E^ArcSin[x]*
(1 + x/Sqrt[1 - x^2]) -
(E^ArcSin[x]*x)/Sqrt[1 - x^2] -
E^ArcSin[x]]
Out[6]=
0
FullSimplify[(1 - x^2)*D[y, {x, 2}] -
x*D[y, {x, 1}] - y]
0
(c) は 上のヒント(a)-(b) を使わず 敢えて 直に ；
Table[D[E^ArcSin[x], {x, k}], {k, 0, 9}]
% /. x -> 0
{E^ArcSin[x], E^ArcSin[x]/Sqrt[1 - x^2],
(E^ArcSin[x]*x)/(1 - x^2)^(3/2) +
E^ArcSin[x]/(1 - x^2),
(3*E^ArcSin[x]*x^2)/(1 - x^2)^(5/2) +
(3*E^ArcSin[x]*x)/(1 - x^2)^2 +
(2*E^ArcSin[x])/(1 - x^2)^(3/2),
(15*E^ArcSin[x]*x^3)/(1 - x^2)^(7/2) +
(15*E^ArcSin[x]*x^2)/(1 - x^2)^3 +
(15*E^ArcSin[x]*x)/(1 - x^2)^(5/2) +
(5*E^ArcSin[x])/(1 - x^2)^2,
(105*E^ArcSin[x]*x^4)/(1 - x^2)^(9/2) +
(105*E^ArcSin[x]*x^3)/(1 - x^2)^4 +
(135*E^ArcSin[x]*x^2)/(1 - x^2)^(7/2) +
(65*E^ArcSin[x]*x)/(1 - x^2)^3 +
(20*E^ArcSin[x])/(1 - x^2)^(5/2),
(945*E^ArcSin[x]*x^5)/(1 - x^2)^(11/2) +
(945*E^ArcSin[x]*x^4)/(1 - x^2)^5 +
(1470*E^ArcSin[x]*x^3)/(1 - x^2)^(9/2) +
(840*E^ArcSin[x]*x^2)/(1 - x^2)^4 +
(435*E^ArcSin[x]*x)/(1 - x^2)^(7/2) +
(85*E^ArcSin[x])/(1 - x^2)^3,
(10395*E^ArcSin[x]*x^6)/(1 - x^2)^
(13/2) + (10395*E^ArcSin[x]*x^5)/
(1 - x^2)^6 + (18900*E^ArcSin[x]*x^4)/
(1 - x^2)^(11/2) +
(11970*E^ArcSin[x]*x^3)/(1 - x^2)^5 +
(8295*E^ArcSin[x]*x^2)/(1 - x^2)^(9/2) +
(2625*E^ArcSin[x]*x)/(1 - x^2)^4 +
(520*E^ArcSin[x])/(1 - x^2)^(7/2),
(135135*E^ArcSin[x]*x^7)/
(1 - x^2)^(15/2) +
(135135*E^ArcSin[x]*x^6)/(1 - x^2)^7 +
(280665*E^ArcSin[x]*x^5)/
(1 - x^2)^(13/2) +
(190575*E^ArcSin[x]*x^4)/(1 - x^2)^6 +
(162225*E^ArcSin[x]*x^3)/
(1 - x^2)^(11/2) +
(65205*E^ArcSin[x]*x^2)/(1 - x^2)^5 +
(22855*E^ArcSin[x]*x)/(1 - x^2)^(9/2) +
(3145*E^ArcSin[x])/(1 - x^2)^4,
(2027025*E^ArcSin[x]*x^8)/
(1 - x^2)^(17/2) +
(2027025*E^ArcSin[x]*x^7)/(1 - x^2)^8 +
(4729725*E^ArcSin[x]*x^6)/
(1 - x^2)^(15/2) +
(3378375*E^ArcSin[x]*x^5)/(1 - x^2)^7 +
(3378375*E^ArcSin[x]*x^4)/
(1 - x^2)^(13/2) +
(1576575*E^ArcSin[x]*x^3)/(1 - x^2)^6 +
(757575*E^ArcSin[x]*x^2)/
(1 - x^2)^(11/2) +
(178425*E^ArcSin[x]*x)/(1 - x^2)^5 +
(26000*E^ArcSin[x])/(1 - x^2)^(9/2)}
{1, 1, 1, 2, 5, 20, 85, 520, 3145, 26000}
参考の為 0の近傍で 下のグラフ達も掲載します。
Table[Normal[Series[E^ArcSin[x],
{x, 0, k}]], {k, 1, 9}]
{1 + x, 1 + x + x^2/2, 1 + x + x^2/2 +
x^3/3, 1 + x + x^2/2 + x^3/3 +
(5*x^4)/24, 1 + x + x^2/2 + x^3/3 +
(5*x^4)/24 + x^5/6, 1 + x + x^2/2 +
x^3/3 + (5*x^4)/24 + x^5/6 +
(17*x^6)/144, 1 + x + x^2/2 + x^3/3 +
(5*x^4)/24 + x^5/6 + (17*x^6)/144 +
(13*x^7)/126, 1 + x + x^2/2 + x^3/3 +
(5*x^4)/24 + x^5/6 + (17*x^6)/144 +
(13*x^7)/126 + (629*x^8)/8064,
1 + x + x^2/2 + x^3/3 + (5*x^4)/24 +
x^5/6 + (17*x^6)/144 + (13*x^7)/126 +
(629*x^8)/8064 + (325*x^9)/4536}
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