![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/spbmathB01a-2.svg\")
<\/p>\n\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\uff0c$$(x^n)’ = n\\, x^{n-1}$$ \u306f\u77e5\u3063\u3066\u3044\u308b\u3068\u3057\u3066\uff0c\u6b63\u306b\u304b\u304e\u3089\u305a\uff0c\u4efb\u610f\u306e\u6574\u6570\u306b\u3064\u3044\u3066\u3082\u540c\u3058\u5fae\u5206\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\uff0c\u3055\u3089\u306b\u306f\u6574\u6570\u3067\u306a\u304f\u3066\u3082\uff0c\u6709\u7406\u6570\uff08\u6574\u6570\u5206\u306e\u6574\u6570\uff09\u4e57\u306e\u3079\u304d\u95a2\u6570\u3067\u3082\uff0c\u6700\u7d42\u7684\u306b\u306f\u4efb\u610f\u306e\u5b9f\u6570 \\(r\\) \u306b\u3064\u3044\u3066
$$(x^r)’ = r\\,x^{r-1}$$
\u3068\u306a\u308b\u3053\u3068\u3082\u308f\u304b\u308b\u3068\u601d\u3046\u3002\uff08\\(r\\) \u304c\u4efb\u610f\u306e\u300c\u5b9f\u6570\u300d\u306e\u5834\u5408\u306f\uff0c\u5f8c\u3067\u8a3c\u660e\u3059\u308b\u3002\uff09<\/p>\n
\u5ff5\u306e\u305f\u3081\u306b\uff0c\u8a3c\u660e\u3002\u3053\u3053\u3067\u306f\u300c\u3079\u304d\u6307\u6570\u300d\\(n\\) \u304c\u6b63\u306e\u6574\u6570\u306e\u5834\u5408\uff0c\u8ca0\u306e\u6574\u6570\u306e\u5834\u5408\uff0c\u6709\u7406\u6570\uff082\u3064\u306e\u6574\u6570\u306e\u5206\u6570\u3067\u8868\u3055\u308c\u308b\u6570\uff09\u306e\u5834\u5408\u306b\u8a3c\u660e\u3059\u308b\u3002<\/p>\n
$$(x^n)’ = n\\, x^{n-1} \\quad (n = 1, 2, 3, \\dots) $$ \u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u8a3c\u660e\u3002<\/p>\n
\u307e\u305a\uff0c\\(n = 1\\) \u306e\u3068\u304d\u306f\uff0c
$$x’ = 1 = 1 \\cdot x^0$$ \u3067\u3042\u308b\u304b\u3089\uff0c\u6210\u308a\u7acb\u3063\u3066\u3044\u308b\u3002
\u6b21\u306b\uff0c\\( n = k\\) \u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068\uff0c
$$ (x^k)’ = k\\, x^{k-1}$$
\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u3092\u4f7f\u3046\u3068\uff0c
\\begin{eqnarray}
\n(x^{k+1})’ &=& (x\\cdot x^{k})’ \\\\
\n&=& x’ \\cdot x^k + x\\cdot (x^k)’ \\\\
\n&=& 1\\cdot x^k + x\\cdot k \\,x^{k-1} \\\\
\n&=& (k+1)\\,x^k
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\\(n = k+1\\) \u306e\u3068\u304d\u306b\u3082\u6210\u7acb\u3059\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u5168\u3066\u306e \\(n = 1, 2, 3, \\dots\\) \u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u305f\u3002<\/p>\n
\u6b21\u306b\uff0c\u8ca0\u306e\u6574\u6570 \\(n = -1, -2, -3, \\cdots\\) \u306b\u3064\u3044\u3066\u306e\u8a3c\u660e\u306f\uff0c\\(n = -m, \\ m = 1, 2, 3, \\cdots\\) \u3068\u304a\u3044\u3066
$$ (x^{-m})’ = (-m)\\,x^{-m-1} \\quad (m = 1, 2, 3, \\cdots)$$ \u3092\u793a\u305b\u3070\u3088\u3044\u3002
\u3053\u308c\u306f\uff0c\u5fae\u5206\u6cd5\u306e\u516c\u5f0f 4. \u3092\u4f7f\u3063\u3066\uff0c
\\begin{eqnarray}
\n(x^{-m})’ &=& \\left\\{\\frac{1}{x^m}\\right\\}’ \\\\
\n&=& -\\frac{(x^m)’}{x^{2m}} \\\\
\n&=& -m\\,x^{(m-1)-2m} \\\\
\n&=& -m\\,x^{-m-1}
\n\\end{eqnarray}<\/p>\n
\u307e\u305f\uff0c\\( n=0\\) \u306e\u3068\u304d\u306f\uff0c
$$(x^0)’ = (1)’ = 0$$ \u306a\u306e\u3067\uff0c\\(n=0\\) \u3068\u3057\u305f\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3063\u3066\u3044\u308b\u3002\u3053\u3053\u307e\u3067\u3092\u307e\u3068\u3081\u308b\u3068\uff0c\u5168\u3066\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064\u3002
$$(x^n)’ = n\\, x^{n-1} \\quad (n = 0, \\pm 1, \\pm 2, \\pm 3, \\dots)$$<\/p>\n
\u6700\u5f8c\u306b\uff0c\u3079\u304d\u6307\u6570\u304c\u4e00\u822c\u306b\u6709\u7406\u6570 \\(r\\) \u306e\u5834\u5408\u306b\u3082
$$(x^r)’ = r\\, x^{r-1}$$ \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u306f\uff0c\u6709\u7406\u6570 \\(r\\) \u304c\u6574\u6570 \\(m, n\\) \u540c\u58eb\u306e\u5272\u308a\u7b97\u3067\u66f8\u3051\u308b\u3053\u3068\u304b\u3089\uff0c\\(\\displaystyle r = \\frac{m}{n}, n > 0\\) \u3068\u3057\u3066
\\begin{eqnarray}
y &=& x^r = x^{\\frac{m}{n}} \\\\
\\therefore \\ y^n &=& x^m \\\\
\\frac{d y^n}{dx}&=& \\frac{d x^m}{dx} \\\\
\\frac{d y^n}{dy} \\frac{dy}{dx}&=&m\\,x^{m-1}\\\\
n\\,y^{n-1} \\frac{dy}{dx}&=& m\\,x^{m-1}
\\end{eqnarray}
\u3057\u305f\u304c\u3063\u3066\uff0c
\\begin{eqnarray}
\\frac{dy}{dx} &=& \\frac{m\\, x^{m-1}}{n\\,y^{n-1}} \\\\
&=& \\frac{m}{n} \\frac{x^{m-1}}{x^{\\frac{m}{n}(n-1)}} \\\\
&=& \\frac{m}{n} \\frac{x^{m-1}}{x^{m-\\frac{m}{n}}} \\\\
&=& \\frac{m}{n} x^{\\frac{m}{n} -1}\\\\
&=& r\\, x^{r-1}
\\end{eqnarray}<\/p>\n
<\/p>\n
\u3059\u308b\u3068\uff0c\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u3088\u308b\u3079\u304d\u95a2\u6570 \\( x^n\\) \u306e\u7dda\u5f62\u7d50\u5408\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u300c\u591a\u9805\u5f0f<\/strong><\/span>\u300d\u3068\u547c\u3070\u308c\u308b\u95a2\u6570 \u591a\u9805\u5f0f\u306e\u5fae\u5206\u306f\uff0c\u5177\u4f53\u7684\u306b\u306f\uff0c\u5fae\u5206\u6cd5\u306e\u516c\u5f0f 1. \u3068 2. \u3092\u4f7f\u3063\u3066\uff0c <\/p>\n","protected":false},"excerpt":{"rendered":" \u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\uff0c$$(x^n)’ = n\\, x^{n-1}$$ \u306f\u77e5\u3063\u3066\u3044\u308b\u3068\u3057\u3066\uff0c\u6b63\u306b\u304b\u304e\u3089\u305a\uff0c\u4efb\u610f\u306e\u6574\u6570\u306b\u3064\u3044\u3066\u3082\u540c\u3058\u5fae\u5206\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\uff0c\u3055\u3089\u306b\u306f\u6574\u6570\u3067\u306a\u304f\u3066\u3082\uff0c\u6709\u7406\u6570\uff08\u6574\u6570\u5206\u306e\u6574\u6570\uff09\u4e57\u306e\u3079\u304d\u95a2\u6570\u3067\u3082\uff0c\u6700\u7d42\u7684\u306b\u306f\u4efb\u610f\u306e\u5b9f\u6570 \\(r\\) \u306b\u3064\u3044\u3066$$(x^r)’ = r\\,x^{r-1}$$\u3068\u306a\u308b\u3053\u3068\u3082\u308f\u304b\u308b\u3068\u601d\u3046\u3002\uff08\\(r\\) \u304c\u4efb\u610f\u306e\u300c\u5b9f\u6570\u300d\u306e\u5834\u5408\u306f\uff0c\u5f8c\u3067\u8a3c\u660e\u3059\u308b\u3002\uff09<\/p>
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0 $$
\u306b\u3064\u3044\u3066\uff0c\u305d\u306e\u5fae\u5206 \\(f'(x)\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u7c21\u5358\u306b\u3067\u304d\u308b\u3057\uff0c\u300c\u591a\u9805\u5f0f\u5206\u306e\u591a\u9805\u5f0f\u300d\u3067\u8868\u3055\u308c\u308b\u300c\u6709\u7406\u95a2\u6570<\/strong><\/span>\u300d\u306e\u5fae\u5206\u3082\u7c21\u5358\u3067\u3059\u306d\u3002\u591a\u9805\u5f0f\u306e\u5e73\u65b9\u6839\u306a\u3069\u3067\u8868\u3055\u308c\u308b\u7121\u7406\u5f0f\u306e\u7a4d\u5206\u3082\u300c\u7121\u7406\u3067\u3059\u300d\u306a\u3093\u3066\u8a00\u308f\u306a\u3044\u3067\uff0c\u591a\u9805\u5f0f\u306e\u3079\u304d\u4e57\u306a\u306e\u3067\uff0c\u3061\u3083\u3093\u3068\u5fae\u5206\u3067\u304d\u307e\u3059\u3088\u306d\u3002<\/p>\n
\\begin{eqnarray}
f'(x) &=& (a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0 )’ \\\\
&=& a_n (x^n)’ + a_{n-1} (x^{n-1})’\u00a0 + \\cdots + a_1 (x)’\u00a0 \\\\
&=& a_n n x^{n-1} + a_{n-1} (n-1) x^{n-2} + \\cdots + a_1
\\end{eqnarray}<\/p>\n