|
|
|
|
|
|
|
|
|
|
|
|
导数计算器
逐步求导
该在线计算器使用常见的求导法则（乘积法则、商法则、链式法则等）来计算任意函数的导数，并展示步骤。它可处理多项式、有理、根式、指数、对数、三角、反三角、双曲及反双曲函数；如有需要，还可在给定点处计算导数值。并且支持计算一阶、二阶、三阶导数，最高到第10阶导数。
相关计算器: 对数求导法计算器, 带步骤的隐函数求导计算器
您的输入
求$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right)$$$。
解答
对 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = \sin{\left(2 x \right)}$$$ 应用乘积法则 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(2 x \right)} + x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)}$$函数$$$\sin{\left(2 x \right)}$$$是两个函数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$和$$$g{\left(x \right)} = 2 x$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。
应用链式法则 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right)$$正弦函数的导数为 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$$x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right)$$返回到原变量：
$$x \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right) = x \cos{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right)$$对 $$$c = 2$$$ 和 $$$f{\left(x \right)} = x$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：
$$x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right) = x \cos{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)} \frac{d}{dx} \left(x\right)$$应用幂法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dx} \left(x\right) = 1$$$：
$$2 x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 2 x \cos{\left(2 x \right)} {\color{red}\left(1\right)} + \sin{\left(2 x \right)} {\color{red}\left(1\right)}$$因此，$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right) = 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}$$$。
答案
$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right) = 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}$$$A