微积分学示例

f (x) = \sin (x) \csc (x)

$f (x) = \sin (x) \csc (x)$

解题步骤 1

使用乘积法则求微分，根据该法则，

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ 等于

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ ，其中

f (x) = \sin (x)

$f (x) = \sin (x)$ 且

g (x) = \csc (x)

$g (x) = \csc (x)$ 。

\sin (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sin (x)]

$\sin (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sin (x)]$

解题步骤 2

\csc (x)

$\csc (x)$ 对

x

$x$ 的导数为

- \csc (x) \cot (x)

$- \csc (x) \cot (x)$ 。

\sin (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sin (x)]

$\sin (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sin (x)]$

解题步骤 3

\sin (x)

$\sin (x)$ 对

x

$x$ 的导数为

\cos (x)

$\cos (x)$ 。

\sin (x) (- \csc (x) \cot (x)) + \csc (x) \cos (x)

$\sin (x) (- \csc (x) \cot (x)) + \csc (x) \cos (x)$

解题步骤 4

化简。

点击获取更多步骤...

解题步骤 4.1

重新排序项。

- \cot (x) \csc (x) \sin (x) + \cos (x) \csc (x)

$- \cot (x) \csc (x) \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2

化简每一项。

点击获取更多步骤...

解题步骤 4.2.1

将

\cot (x)

$\cot (x)$ 重写为正弦和余弦形式。

- \frac{\cos (x)}{\sin (x)} \csc (x) \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} \csc (x) \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.2

将

\csc (x)

$\csc (x)$ 重写为正弦和余弦形式。

- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.3

乘以

- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ 。

点击获取更多步骤...

解题步骤 4.2.3.1

将

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ 乘以

\frac{\cos (x)}{\sin (x)}

$\frac{\cos (x)}{\sin (x)}$ 。

- \frac{\cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.3.2

对

\sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

- \frac{\cos (x)}{\sin^{1} (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{1} (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.3.3

对

\sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

- \frac{\cos (x)}{\sin^{1} (x) \sin^{1} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{1} (x) \sin^{1} (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.3.4

使用幂法则

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 合并指数。

- \frac{\cos (x)}{{\sin (x)}^{1 + 1}} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{{\sin (x)}^{1 + 1}} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.3.5

将

1

$1$ 和

1

$1$ 相加。

- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.4

约去

\sin (x)

$\sin (x)$ 的公因数。

点击获取更多步骤...

解题步骤 4.2.4.1

将

- \frac{\cos (x)}{\sin^{2} (x)}

$- \frac{\cos (x)}{\sin^{2} (x)}$ 中前置负号移到分子中。

\frac{- \cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.4.2

从

\sin^{2} (x)

$\sin^{2} (x)$ 中分解出因数

\sin (x)

$\sin (x)$ 。

\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.4.3

约去公因数。

\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

解题步骤 4.2.4.4

重写表达式。

\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)$

\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)$

解题步骤 4.2.5

将负号移到分数的前面。

- \frac{\cos (x)}{\sin (x)} + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} + \cos (x) \csc (x)$

解题步骤 4.2.6

将

\csc (x)

$\csc (x)$ 重写为正弦和余弦形式。

- \frac{\cos (x)}{\sin (x)} + \cos (x) \frac{1}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \cos (x) \frac{1}{\sin (x)}$

解题步骤 4.2.7

组合

\cos (x)

$\cos (x)$ 和

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ 。

- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}$

- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}$

解题步骤 4.3

将

- \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)}$ 和

\frac{\cos (x)}{\sin (x)}

$\frac{\cos (x)}{\sin (x)}$ 相加。

0

$0$

0

$0$

微积分学 示例

微积分学示例