基本関数の微分

基本関数の微分

(1)定数

\[ (c)'=0 \]

(2)べき関数

\[ \left(x^{\alpha}\right)'=\alpha x^{\alpha-1} \]

(3)指数関数

\[ \left(e^{x}\right)'=e^{x} \]

(4)指数関数

\[ \left(a^{x}\right)'=a^{x}\log a \]

(5)自然対数関数

\[ \left(\log x\right)'=\frac{1}{x} \]

(6)対数関数

\[ \left(\log_{a}x\right)'=\frac{1}{x\log a} \]

(7)自然対数関数

\[ \left(\log\left|x\right|\right)'=\frac{1}{\left|x\right|} \]

(8)対数関数

\[ \left(\log_{a}\left|x\right|\right)'=\frac{1}{\left|x\right|\log a} \]

(1)

\begin{align*} (c)' & =\lim_{\Delta x\rightarrow0}\frac{c-c}{\Delta x}\\ & =0 \end{align*}

(2)

\begin{align*} \left(x^{\alpha}\right)' & =\lim_{\Delta x\rightarrow0}\frac{\left(x+\Delta x\right)^{\alpha}-x^{\alpha}}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left\{ \sum_{k=0}^{\infty}C(\alpha,k)x^{\alpha-k}\Delta x^{k}-x^{\alpha}\right\} \\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left\{ \sum_{k=1}^{\infty}C(\alpha,k)x^{\alpha-k}\Delta x^{k}\right\} \\ & =\lim_{\Delta x\rightarrow0}\left\{ C(\alpha,1)x^{\alpha-1}+\sum_{k=2}^{\infty}C(\alpha,k)x^{\alpha-k}\Delta x^{k-1}\right\} \\ & =\alpha x^{\alpha-1} \end{align*}

(3)

(4)より、

\begin{align*} \left(e^{x}\right)' & =e^{x}\log e\\ & =e^{x} \end{align*}

(4)

\begin{align*} \left(a^{x}\right)' & =\lim_{\Delta x\rightarrow0}\frac{a^{x+\Delta x}-a^{x}}{\Delta x}\\ & =a^{x}\lim_{\Delta x\rightarrow0}\frac{a^{\Delta x}-1}{\Delta x}\\ & =a^{x}\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left\{ \left(1+\Delta x\right)^{\frac{1}{\Delta x}\log a\cdot\Delta x}-1\right\} \\ & =a^{x}\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left\{ \left(1+\Delta x\right)^{\log a}-1\right\} \\ & =a^{x}\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left\{ \left(1+\Delta x\log a\right)-1\right\} \\ & =a^{x}\log a \end{align*}

(5)

\begin{align*} \left(\log x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\log(x+\Delta x)-\log x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\log\frac{x+\Delta x}{x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\log\left(1+\frac{\Delta x}{x}\right)\\ & =\lim_{\Delta x\rightarrow0}\log\left(1+\frac{\Delta x}{x}\right)^{\frac{x}{\Delta x}\frac{1}{x}}\\ & =\log e^{\frac{1}{x}}\\ & =\frac{1}{x} \end{align*}

(6)

\begin{align*} \left(\log_{a}x\right)' & =\frac{1}{\log a}\left(\log x\right)'\\ & =\frac{1}{x\log a} \end{align*}

(7)

\begin{align*} \left(\log\left|x\right|\right)' & =\left(\log\sgn x\right)'\\ & =\lim_{\Delta x\rightarrow0}\frac{\log(\sgn x+\Delta x)-\log\sgn x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\log\frac{\sgn x+\Delta x}{\sgn x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\log\left(1+\frac{\Delta x}{\sgn x}\right)\\ & =\lim_{\Delta x\rightarrow0}\log\left(1+\frac{\Delta x}{\sgn x}\right)^{\frac{\sgn x}{\Delta x}\frac{1}{\sgn x}}\\ & =\log e^{\frac{1}{\sgn x}}\\ & =\frac{1}{\sgn x}\\ & =\frac{1}{\left|x\right|} \end{align*}

(8)

\begin{align*} \left(\log_{a}\left|x\right|\right)' & =\frac{1}{\log a}\left(\log\left|x\right|\right)'\\ & =\frac{1}{\left|x\right|\log a} \end{align*}

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基本関数の微分

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