数学のブログ

微分法 導関数とその計算 累乗、平方根、和、積、商、合成関数

微分積分演習〈理工系の数学入門コース/演習 新装版〉 (和達 三樹(著)、十河 清(著)、岩波書店)の第3章(微分法)、3-1(導関数とその計算)、問題1の解答を求めてみる。

1

d dx x 3 ( 1 + x 2 ) = 3 x 2 + 5 x 4 = x 2 ( 3 + 5 x 2 )

2

d dx x 2 + 1 = 2 x 2 x 2 + 1 = x x 2 + 1

3

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

4

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

コード(Wolfram Language, Jupyter)

f[x_] := x^3(1+x^2)

f'[x]
Output
% == x^2(3+5x^2)
Output
Simplify[%]
Output
Plot[{f[x], f'[x]}, {x, -5, 5}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
f[x_] := Sqrt[x^2+1]

f'[x] == x / Sqrt[x^2+1]
Output
Plot[{f[x], f'[x]}, {x, -5, 5}, PlotRange -> {-5, 5}, PlotLegends -> "Expressions"]
Output
f[x_] := 1/(x(x+1))

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

f'[x] == -(2x-a-b)/(2(x-a)(x-b)Sqrt[(x-a)(x-b)])
Output
Simplify[%]
Output
f[x_] := 1/Sqrt[(x-1)(x-2)]

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