$$$\ln^{3}\left(u\right)$$$の導関数
入力内容
$$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right)$$$ を求めよ。
解答
関数$$$\ln^{3}\left(u\right)$$$は、2つの関数$$$f{\left(v \right)} = v^{3}$$$と$$$g{\left(u \right)} = \ln\left(u\right)$$$の合成$$$f{\left(g{\left(u \right)} \right)}$$$である。
連鎖律 $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$ を適用する:
$${\color{red}\left(\frac{d}{du} \left(\ln^{3}\left(u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{3}\right) \frac{d}{du} \left(\ln\left(u\right)\right)\right)}$$冪法則 $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ を $$$n = 3$$$ に対して適用する:
$${\color{red}\left(\frac{d}{dv} \left(v^{3}\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = {\color{red}\left(3 v^{2}\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$元の変数に戻す:
$$3 {\color{red}\left(v\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right) = 3 {\color{red}\left(\ln\left(u\right)\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right)$$自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$$3 \ln^{2}\left(u\right) {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} = 3 \ln^{2}\left(u\right) {\color{red}\left(\frac{1}{u}\right)}$$したがって、$$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$。
解答
$$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$A
Please try a new game Rotatly