$$$\ln\left(x\right) \sin{\left(9 x \right)}$$$の導関数

$$$H{\left(x \right)} = \ln\left(x\right) \sin{\left(9 x \right)}$$$ とする。

両辺の対数を取る: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(\ln\left(x\right) \sin{\left(9 x \right)}\right)$$$.

対数の性質を用いて右辺を書き換えよ: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)$$$。

方程式の両辺をそれぞれ微分せよ: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right)$$$

方程式の左辺を微分せよ。

関数$$$\ln\left(H{\left(x \right)}\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = H{\left(x \right)}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$

自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$

元の変数に戻す:

$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$

したがって、$$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$。

方程式の右辺を微分する。

和/差の導関数は、導関数の和/差である:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)\right)}$$

関数$$$\ln\left(\ln\left(x\right)\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = \ln\left(x\right)$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)$$

自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)$$

元の変数に戻す:

$$\frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{{\color{red}\left(\ln\left(x\right)\right)}}$$

関数$$$\ln\left(\sin{\left(9 x \right)}\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = \sin{\left(9 x \right)}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)\right)} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sin{\left(9 x \right)}\right)\right)} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sin{\left(9 x \right)}\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sin{\left(9 x \right)}\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

元の変数に戻す:

$$\frac{\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)}{{\color{red}\left(u\right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)}{{\color{red}\left(\sin{\left(9 x \right)}\right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

関数$$$\sin{\left(9 x \right)}$$$は、2つの関数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$と$$$g{\left(x \right)} = 9 x$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{{\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(9 x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

正弦関数の導関数は$$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$$\frac{{\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{{\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

元の変数に戻す:

$$\frac{\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\cos{\left({\color{red}\left(9 x\right)} \right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

定数倍の法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ を $$$c = 9$$$ と $$$f{\left(x \right)} = x$$$ に対して適用します:

$$\frac{\cos{\left(9 x \right)} {\color{red}\left(\frac{d}{dx} \left(9 x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\cos{\left(9 x \right)} {\color{red}\left(9 \frac{d}{dx} \left(x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

自然対数の導関数は $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:

$$\frac{9 \cos{\left(9 x \right)} \frac{d}{dx} \left(x\right)}{\sin{\left(9 x \right)}} + \frac{{\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{\ln\left(x\right)} = \frac{9 \cos{\left(9 x \right)} \frac{d}{dx} \left(x\right)}{\sin{\left(9 x \right)}} + \frac{{\color{red}\left(\frac{1}{x}\right)}}{\ln\left(x\right)}$$

$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\frac{9 \cos{\left(9 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)} = \frac{9 \cos{\left(9 x \right)} {\color{red}\left(1\right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$

簡単化せよ:

$$\frac{9 \cos{\left(9 x \right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)} = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$

したがって、$$$\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right) = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$$。

したがって、$$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$$。

したがって、$$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(\frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}\right) H{\left(x \right)} = 9 \ln\left(x\right) \cos{\left(9 x \right)} + \frac{\sin{\left(9 x \right)}}{x}$$$。