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$$$x^{7 x}$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(x^{7 x}\right)$$$。
解答
設 $$$H{\left(x \right)} = x^{7 x}$$$。
對等式兩邊取對數：$$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{7 x}\right)$$$。
利用對數的性質改寫等式右邊：$$$\ln\left(H{\left(x \right)}\right) = 7 x \ln\left(x\right)$$$。
將等式兩邊分別微分：$$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(7 x \ln\left(x\right)\right)$$$。
對等式左邊求導數。
函數 $$$\ln\left(H{\left(x \right)}\right)$$$ 是兩個函數 $$$f{\left(u \right)} = \ln\left(u\right)$$$ 與 $$$g{\left(x \right)} = H{\left(x \right)}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$自然對數的導數為 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$返回原變數：
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$因此，$$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$。
對等式右邊求導。
套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = 7$$$ 與 $$$f{\left(x \right)} = x \ln\left(x\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(7 x \ln\left(x\right)\right)\right)} = {\color{red}\left(7 \frac{d}{dx} \left(x \ln\left(x\right)\right)\right)}$$將乘積法則 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ 應用於 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = \ln\left(x\right)$$$：
$$7 {\color{red}\left(\frac{d}{dx} \left(x \ln\left(x\right)\right)\right)} = 7 {\color{red}\left(\frac{d}{dx} \left(x\right) \ln\left(x\right) + x \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$自然對數的導數為 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$：
$$7 x {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + 7 \ln\left(x\right) \frac{d}{dx} \left(x\right) = 7 x {\color{red}\left(\frac{1}{x}\right)} + 7 \ln\left(x\right) \frac{d}{dx} \left(x\right)$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$7 \ln\left(x\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 7 = 7 \ln\left(x\right) {\color{red}\left(1\right)} + 7$$因此，$$$\frac{d}{dx} \left(7 x \ln\left(x\right)\right) = 7 \ln\left(x\right) + 7$$$。
因此，$$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = 7 \ln\left(x\right) + 7$$$。
因此，$$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(7 \ln\left(x\right) + 7\right) H{\left(x \right)} = 7 x^{7 x} \left(\ln\left(x\right) + 1\right)$$$。
答案
$$$\frac{d}{dx} \left(x^{7 x}\right) = 7 x^{7 x} \left(\ln\left(x\right) + 1\right)$$$A