解答 - 导数

使用链式法则计算正切函数的导数。

$$\frac{d}{dx}\left[\tan \left(x^2\right)\right]=\frac{1}{\cos {\left(x^2\right)}^2}\times \frac{d}{dx}\left[x^2\right]$$

为链式法则分解函数。

$$\frac{d}{dx}\left[\tan \left(x^2\right)\right]=\frac{d}{dx}\left[\tan \left(x\right)\right]\times \frac{d}{dx}\left[x^2\right]$$

使用链式法则计算正切函数的导数。

$$\frac{d}{dx}\left[\tan \left(x^2\right)\right]=\frac{1}{\cos {\left(x^2\right)}^2}\times \frac{d}{dx}\left[x^2\right]$$

角的正切等于角的正弦除以角的余弦。

$$\frac{d}{dx}\left[\tan \left(x\right)\right]=\frac{d}{dx}\left[\frac{\sin \left(x\right)}{\cos \left(x\right)}\right]$$

计算分数的导数。

$$\frac{d}{dx}\left[\frac{\sin \left(x\right)}{\cos \left(x\right)}\right]=\frac{\frac{d}{dx}\left[\sin \left(x\right)\right]\times \cos \left(x\right)-\sin \left(x\right)\times \frac{d}{dx}\left[\cos \left(x\right)\right]}{\cos {\left(x\right)}^2}$$

计算正弦函数的导数。

$$\frac{\frac{d}{dx}\left[\sin \left(x\right)\right]\times \cos \left(x\right)-\sin \left(x\right)\times \frac{d}{dx}\left[\cos \left(x\right)\right]}{\cos {\left(x\right)}^2}=\frac{\cos \left(x\right)\times \cos \left(x\right)-\sin \left(x\right)\times \frac{d}{dx}\left[\cos \left(x\right)\right]}{\cos {\left(x\right)}^2}$$

计算余弦函数的导数。

$$\frac{\cos \left(x\right)\times \cos \left(x\right)-\sin \left(x\right)\times \frac{d}{dx}\left[\cos \left(x\right)\right]}{\cos {\left(x\right)}^2}=\frac{\cos \left(x\right)\times \cos \left(x\right)-\sin \left(x\right)\times \left(-\sin \left(x\right)\right)}{\cos {\left(x\right)}^2}$$

取消两个负号，结果为正值。

$$\frac{\cos \left(x\right)\times \cos \left(x\right)-\sin \left(x\right)\times \left(-\sin \left(x\right)\right)}{\cos {\left(x\right)}^2}=\frac{\cos \left(x\right)\times \cos \left(x\right)+\sin \left(x\right)\times \sin \left(x\right)}{\cos {\left(x\right)}^2}$$

将相同的数相乘。

$$\frac{\cos \left(x\right)\times \cos \left(x\right)+\sin \left(x\right)\times \sin \left(x\right)}{\cos {\left(x\right)}^2}=\frac{\cos {\left(x\right)}^2+\sin \left(x\right)\times \sin \left(x\right)}{\cos {\left(x\right)}^2}$$

将相同的数相乘。

$$\frac{\cos {\left(x\right)}^2+\sin \left(x\right)\times \sin \left(x\right)}{\cos {\left(x\right)}^2}=\frac{\cos {\left(x\right)}^2+\sin {\left(x\right)}^2}{\cos {\left(x\right)}^2}$$

一个角的余弦的平方加上同一个角的正弦的平方总是等于一。

$$\frac{\cos {\left(x\right)}^2+\sin {\left(x\right)}^2}{\cos {\left(x\right)}^2}=\frac{1}{\cos {\left(x\right)}^2}$$

将变量重新替换回函数。

$$\frac{1}{\cos {\left(x\right)}^2}\times \frac{d}{dx}\left[x^2\right]=\frac{1}{\cos {\left(x^2\right)}^2}\times \frac{d}{dx}\left[x^2\right]$$