$$$5 x^{x}$$$ 的導數

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 $$$c = 5$$$ 與 $$$f{\left(x \right)} = x^{x}$$$：

$${\color{red}\left(\frac{d}{dx} \left(5 x^{x}\right)\right)} = {\color{red}\left(5 \frac{d}{dx} \left(x^{x}\right)\right)}$$

使用公式 $$$f^{g{\left(x \right)}}{\left(x \right)} = e^{g{\left(x \right)} \ln\left(f{\left(x \right)}\right)}$$$，以 $$$f{\left(x \right)} = x$$$ 與 $$$g{\left(x \right)} = x$$$ 來重寫複雜表達式：

$$5 {\color{red}\left(\frac{d}{dx} \left(x^{x}\right)\right)} = 5 {\color{red}\left(\frac{d}{dx} \left(e^{x \ln\left(x\right)}\right)\right)}$$

函數 $$$e^{x \ln\left(x\right)}$$$ 是兩個函數 $$$f{\left(u \right)} = e^{u}$$$ 與 $$$g{\left(x \right)} = x \ln\left(x\right)$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

$$5 {\color{red}\left(\frac{d}{dx} \left(e^{x \ln\left(x\right)}\right)\right)} = 5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(x \ln\left(x\right)\right)\right)}$$

指數函數的導數為 $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$：

$$5 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(x \ln\left(x\right)\right) = 5 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(x \ln\left(x\right)\right)$$

返回原變數：

$$5 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(x \ln\left(x\right)\right) = 5 e^{{\color{red}\left(x \ln\left(x\right)\right)}} \frac{d}{dx} \left(x \ln\left(x\right)\right) = 5 x^{x} \frac{d}{dx} \left(x \ln\left(x\right)\right)$$

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 $$$g{\left(x \right)} = \ln\left(x\right)$$$：

$$5 x^{x} {\color{red}\left(\frac{d}{dx} \left(x \ln\left(x\right)\right)\right)} = 5 x^{x} {\color{red}\left(\frac{d}{dx} \left(x\right) \ln\left(x\right) + x \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$

自然對數的導數為 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$：

$$5 x^{x} \left(x {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \ln\left(x\right) \frac{d}{dx} \left(x\right)\right) = 5 x^{x} \left(x {\color{red}\left(\frac{1}{x}\right)} + \ln\left(x\right) \frac{d}{dx} \left(x\right)\right)$$

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：

$$5 x^{x} \left(\ln\left(x\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 1\right) = 5 x^{x} \left(\ln\left(x\right) {\color{red}\left(1\right)} + 1\right)$$

因此，$$$\frac{d}{dx} \left(5 x^{x}\right) = 5 x^{x} \left(\ln\left(x\right) + 1\right)$$$。