x^2 - 48*x 的導數

此計算器將求出 $$$x^{2} - 48 x$$$ 的導數，並顯示步驟。

您的輸入

求$$$\frac{d}{dx} \left(x^{2} - 48 x\right)$$$。

解答

和/差的導數等於導數的和/差：

$${\color{red}\left(\frac{d}{dx} \left(x^{2} - 48 x\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) - \frac{d}{dx} \left(48 x\right)\right)}$$

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 2$$$：

$${\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(48 x\right) = {\color{red}\left(2 x\right)} - \frac{d}{dx} \left(48 x\right)$$

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 $$$c = 48$$$ 與 $$$f{\left(x \right)} = x$$$：

$$2 x - {\color{red}\left(\frac{d}{dx} \left(48 x\right)\right)} = 2 x - {\color{red}\left(48 \frac{d}{dx} \left(x\right)\right)}$$

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

$$2 x - 48 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 2 x - 48 {\color{red}\left(1\right)}$$

因此，$$$\frac{d}{dx} \left(x^{2} - 48 x\right) = 2 x - 48$$$。

答案

$$$\frac{d}{dx} \left(x^{2} - 48 x\right) = 2 x - 48$$$A

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