数学のブログ

自然法則の微分方程式 微積分の予備知識 導関数、累乗、積、商、三角関数(正弦と余弦)、指数関数、対数関数、合成関数の微分

微分方程式演習〈理工系の数学入門コース/演習 新装版〉 (和達 三樹(著)、矢嶋 徹(著)、岩波書店)の第1章(自然法則の微分方程式)、1-1(微積分の予備知識)、問題1の解答を求めてみる。

1

d dx f ( x ) = 4 x 3 + 3 x 2 + 2 x + 1

2

d dx x a = a x a - 1

3

d dx ( a x + b ) c = a c ( a x + b ) c - 1

4

d dx cos a x sin b x = - a ( sin a x ) sin b x + b cos a x cos b x

5

d dx x log x = log x + x · 1 x = log x + 1

6

d dx a x = d dx e x log a = e x log a log a = a x log a

7

d dx tan x = 1 cos 2 x

8

d dx x x 2 + 1 = x 2 + 1 - x 2 x ( x 2 + 1 ) 2 = 1 - x 2 x 2 + 1

9

d dx e x 2 = 2 x e x 2

10

d dx cos ( log | x | ) = - 1 x sin log | x |

11

d dx log ( log | x | ) = 1 x log | x |

12

d dx e - i a x = - i a e - a x

13

d dx x x = e x log x = ( log x + 1 ) e x log x = x x ( log x + 1 )

14

d dx x log | x | = e ( log | x | ) 2 = 2 log | x | x e ( log | x | ) 2 = 2 log | x | x x log | x | = 2 x log | x | - 1 log | x |

コード(Wolfram Language, Jupyter)

f[x_] := x^4+x^3+x^2+x+1;
f'[x]
Output
TraditionalForm[%]
Output
Plot[{f[x], f'[x]}, {x, -5, 5}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
f[x_] := x^a

f'[x]
Output
D[(a x+b)^c, x]
Output
D[Cos[a x] Sin[b x], x]
Output
f[x_] := x Log[x]

f'[x]
Output
Plot[{f[x], f'[x]}, {x, 0, 10}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
D[a^x, x]
Output
D[Tan[x], x]
Output
% == 1 / Cos[x]^2
Output
Plot[{Tan[x], 1 / Cos[x]^2},
     {x, -5, 5},
     PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
D[x / (x^2+1), x]
Output
Simplify[%]
Output
f[x_] := x / (x^2 + 1)

Plot[{f[x], f'[x]}, {x, -5, 5},
     PlotRange -> {-5, 5},
     PlotLegends -> "Expressions"
]
Output
f[x_] := Exp[x^2]

f'[x]
Output
Plot[{f[x], f'[x]}, {x, -5, 5},
     PlotRange -> {-5, 5},
     PlotLegends -> "Expressions"
]
Output
f[x_] := Cos[Log[x]]

f'[x]
Output
Plot[{f[x], f'[x]}, {x, 0, 10},
     PlotRange -> {-10, 10},
     PlotLegends -> "Expressions"
]
Output
f[x_] := Log[Log[x]]

f'[x]
Output
Plot[{f[x], f'[x]}, {x, 0, 10},
     PlotRange -> {-10, 10},
     PlotLegends -> "Expressions"
]
Output
f[x_, a_] := Exp[-I a x]

D[f[x, a], x]
Output
ComplexPlot[f[x, 1], {x, -2-2I, 2+2I}]
Output
ComplexPlot[D[f[x, 1], x], {x, -2-2I, 2+2I}]
Output
D[x^x,x]
Output
Plot[{x^x, x^x (1+Log[x])}, {x, 0, 10}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
D[x^Log[x], x]
Output
f[x_] := x^Log[x]

f'[x]
Output
Plot[{f[x], f'[x]}, {x, 0, 10}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output