x + sin(x)的导数

该计算器将求$$$x + \sin{\left(x \right)}$$$的导数，并显示步骤。

您的输入

求$$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right)$$$。

解答

和/差的导数等于导数的和/差：

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

正弦函数的导数为 $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x\right)$$

应用幂法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dx} \left(x\right) = 1$$$：

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

因此，$$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 1$$$。

答案

$$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 1$$$A

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