带步骤的在线导数计算器

在线计算器将使用常见的微分规则(乘积规则、商规则、链规则等)计算任何函数的导数,并显示步骤。它可以处理多项式、有理、无理、指数、对数、三角、反三角、双曲和反双曲函数。此外,如果需要,它将评估给定点的导数。它还支持计算一阶、二阶和三阶导数,最多 10 个。

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您的输入

查找$$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$$$

解决方案

应用带有$$$f{\left(x \right)} = x$$$$$$g{\left(x \right)} = \sin{\left(x \right)}$$$的乘积规则$$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$

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

应用具有$$$n = 1$$$的幂规则$$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ ,换句话说, $$$\frac{d}{dx} \left(x\right) = 1$$$

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

正弦的导数是$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$

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

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

回答

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