5 - 8*x^4 的導數

此計算器將求出 $$$5 - 8 x^{4}$$$ 的導數，並顯示步驟。

您的輸入

求$$$\frac{d}{dx} \left(5 - 8 x^{4}\right)$$$。

解答

和/差的導數等於導數的和/差：

$${\color{red}\left(\frac{d}{dx} \left(5 - 8 x^{4}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(5\right) - \frac{d}{dx} \left(8 x^{4}\right)\right)}$$

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 $$$c = 8$$$ 與 $$$f{\left(x \right)} = x^{4}$$$：

$$- {\color{red}\left(\frac{d}{dx} \left(8 x^{4}\right)\right)} + \frac{d}{dx} \left(5\right) = - {\color{red}\left(8 \frac{d}{dx} \left(x^{4}\right)\right)} + \frac{d}{dx} \left(5\right)$$

常數的導數為$$$0$$$：

$${\color{red}\left(\frac{d}{dx} \left(5\right)\right)} - 8 \frac{d}{dx} \left(x^{4}\right) = {\color{red}\left(0\right)} - 8 \frac{d}{dx} \left(x^{4}\right)$$

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 4$$$：

$$- 8 {\color{red}\left(\frac{d}{dx} \left(x^{4}\right)\right)} = - 8 {\color{red}\left(4 x^{3}\right)}$$

因此，$$$\frac{d}{dx} \left(5 - 8 x^{4}\right) = - 32 x^{3}$$$。

答案

$$$\frac{d}{dx} \left(5 - 8 x^{4}\right) = - 32 x^{3}$$$A