$$$\sin{\left(5 \theta \right)}$$$的导数

求$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)$$$。

函数$$$\sin{\left(5 \theta \right)}$$$是两个函数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$和$$$g{\left(\theta \right)} = 5 \theta$$$的复合$$$f{\left(g{\left(\theta \right)} \right)}$$$。

应用链式法则 $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$：

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

返回到原变量：

对 $$$c = 5$$$ 和 $$$f{\left(\theta \right)} = \theta$$$ 应用常数倍法则 $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$：

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

因此，$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$。

$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$A