微积分学示例

$y = \sqrt[5]{\frac{x}{x^{2} - 1}}$

解题步骤 1

求一阶导数。

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解题步骤 1.1

使用 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将 $\sqrt[5]{\frac{x}{x^{2} - 1}}$ 重写成 ${(\frac{x}{x^{2} - 1})}^{\frac{1}{5}}$ 。

$\frac{d}{d x} [{(\frac{x}{x^{2} - 1})}^{\frac{1}{5}}]$

解题步骤 1.2

使用链式法则求微分，根据该法则， $\frac{d}{d x} [f (g (x))]$ 等于 $f' (g (x)) g' (x)$ ，其中 $f (x) = x^{\frac{1}{5}}$ 且 $g (x) = \frac{x}{x^{2} - 1}$ 。

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解题步骤 1.2.1

要使用链式法则，请将 $u$ 设为 $\frac{x}{x^{2} - 1}$ 。

$\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.2.2

使用幂法则求微分，根据该法则， $\frac{d}{d u} [u^{n}]$ 等于 $n u^{n - 1}$ ，其中 $n = \frac{1}{5}$ 。

$\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.2.3

使用 $\frac{x}{x^{2} - 1}$ 替换所有出现的 $u$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} - 1} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.3

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{5}{5}$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.4

组合 $- 1$ 和 $\frac{5}{5}$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.5

在公分母上合并分子。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1 - 1 \cdot 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.6

化简分子。

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解题步骤 1.6.1

将 $- 1$ 乘以 $5$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1 - 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.6.2

从 $1$ 中减去 $5$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{- 4}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.7

将负号移到分数的前面。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

解题步骤 1.8

使用除法定则求微分，根据该法则， $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ 等于 $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ ，其中 $f (x) = x$ 且 $g (x) = x^{2} - 1$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{(x^{2} - 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9

求微分。

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解题步骤 1.9.1

使用幂法则求微分，根据该法则， $\frac{d}{d x} [x^{n}]$ 等于 $n x^{n - 1}$ ，其中 $n = 1$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{(x^{2} - 1) \cdot 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.2

将 $x^{2} - 1$ 乘以 $1$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.3

根据加法法则， $x^{2} - 1$ 对 $x$ 的导数是 $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.4

使用幂法则求微分，根据该法则， $\frac{d}{d x} [x^{n}]$ 等于 $n x^{n - 1}$ ，其中 $n = 2$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.5

因为 $- 1$ 对于 $x$ 是常数，所以 $- 1$ 对 $x$ 的导数为 $0$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (2 x + 0)}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.6

化简表达式。

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解题步骤 1.9.6.1

将 $2 x$ 和 $0$ 相加。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (2 x)}{{(x^{2} - 1)}^{2}}$

解题步骤 1.9.6.2

将 $2$ 乘以 $- 1$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 x \cdot x}{{(x^{2} - 1)}^{2}}$

解题步骤 1.10

对 $x$ 进行 $1$ 次方运算。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 (x^{1} x)}{{(x^{2} - 1)}^{2}}$

解题步骤 1.11

对 $x$ 进行 $1$ 次方运算。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 (x^{1} x^{1})}{{(x^{2} - 1)}^{2}}$

解题步骤 1.12

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 x^{1 + 1}}{{(x^{2} - 1)}^{2}}$

解题步骤 1.13

将 $1$ 和 $1$ 相加。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 x^{2}}{{(x^{2} - 1)}^{2}}$

解题步骤 1.14

从 $x^{2}$ 中减去 $2 x^{2}$ 。

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{- x^{2} - 1}{{(x^{2} - 1)}^{2}}$

解题步骤 1.15

将 $\frac{- x^{2} - 1}{{(x^{2} - 1)}^{2}}$ 乘以 $\frac{1}{5}$ 。

$\frac{- x^{2} - 1}{{(x^{2} - 1)}^{2} \cdot 5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}}$

解题步骤 1.16

将 $5$ 移到 ${(x^{2} - 1)}^{2}$ 的左侧。

$\frac{- x^{2} - 1}{5 \cdot {(x^{2} - 1)}^{2}} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}}$

解题步骤 1.17

化简。

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解题步骤 1.17.1

通过将底数重写为其倒数的方式改变指数的符号。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{2}} {(\frac{x^{2} - 1}{x})}^{\frac{4}{5}}$

解题步骤 1.17.2

对 $\frac{x^{2} - 1}{x}$ 运用乘积法则。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{2}} \cdot \frac{{(x^{2} - 1)}^{\frac{4}{5}}}{x^{\frac{4}{5}}}$

解题步骤 1.17.3

合并项。

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解题步骤 1.17.3.1

将 $\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{2}}$ 乘以 $\frac{{(x^{2} - 1)}^{\frac{4}{5}}}{x^{\frac{4}{5}}}$ 。

$\frac{(- x^{2} - 1) {(x^{2} - 1)}^{\frac{4}{5}}}{5 {(x^{2} - 1)}^{2} x^{\frac{4}{5}}}$

解题步骤 1.17.3.2

使用负指数规则 $b^{n} = \frac{1}{b^{- n}}$ 将 ${(x^{2} - 1)}^{\frac{4}{5}}$ 移动到分母。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{2} x^{\frac{4}{5}} {(x^{2} - 1)}^{- \frac{4}{5}}}$

解题步骤 1.17.3.3

通过指数相加将 ${(x^{2} - 1)}^{2}$ 乘以 ${(x^{2} - 1)}^{- \frac{4}{5}}$ 。

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解题步骤 1.17.3.3.1

移动 ${(x^{2} - 1)}^{- \frac{4}{5}}$ 。

$\frac{- x^{2} - 1}{5 ({(x^{2} - 1)}^{- \frac{4}{5}} {(x^{2} - 1)}^{2}) x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{- \frac{4}{5} + 2} x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.3

要将 $2$ 写成带有公分母的分数，请乘以 $\frac{5}{5}$ 。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{- \frac{4}{5} + 2 \cdot \frac{5}{5}} x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.4

组合 $2$ 和 $\frac{5}{5}$ 。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{- \frac{4}{5} + \frac{2 \cdot 5}{5}} x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.5

在公分母上合并分子。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{\frac{- 4 + 2 \cdot 5}{5}} x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.6

化简分子。

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解题步骤 1.17.3.3.6.1

将 $2$ 乘以 $5$ 。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{\frac{- 4 + 10}{5}} x^{\frac{4}{5}}}$

解题步骤 1.17.3.3.6.2

将 $- 4$ 和 $10$ 相加。

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{\frac{6}{5}} x^{\frac{4}{5}}}$

解题步骤 1.17.4

重新排序项。

$\frac{- x^{2} - 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 1.17.5

从 $- x^{2}$ 中分解出因数 $- 1$ 。

$\frac{- (x^{2}) - 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 1.17.6

将 $- 1$ 重写为 $- 1 (1)$ 。

$\frac{- (x^{2}) - 1 (1)}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 1.17.7

从 $- (x^{2}) - 1 (1)$ 中分解出因数 $- 1$ 。

$\frac{- (x^{2} + 1)}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 1.17.8

将 $- (x^{2} + 1)$ 重写为 $- 1 (x^{2} + 1)$ 。

$\frac{- 1 (x^{2} + 1)}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 1.17.9

将负号移到分数的前面。

$f' (x) = - \frac{x^{2} + 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

解题步骤 2

求二阶导数。

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解题步骤 2.1

因为 $- \frac{1}{5}$ 对于 $x$ 是常数，所以 $- \frac{x^{2} + 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$ 对 $x$ 的导数是 $- \frac{1}{5} \frac{d}{d x} [\frac{x^{2} + 1}{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}]$ 。

$- \frac{1}{5} \frac{d}{d x} [\frac{x^{2} + 1}{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}]$

解题步骤 2.2

使用除法定则求微分，根据该法则， $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ 等于 $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ ，其中 $f (x) = x^{2} + 1$ 且 $g (x) = x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 。

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{2} + 1] - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.3

求微分。

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解题步骤 2.3.1

根据加法法则， $x^{2} + 1$ 对 $x$ 的导数是 $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ 。

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.3.2

使用幂法则求微分，根据该法则， $\frac{d}{d x} [x^{n}]$ 等于 $n x^{n - 1}$ ，其中 $n = 2$ 。

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.3.3

因为 $1$ 对于 $x$ 是常数，所以 $1$ 对 $x$ 的导数为 $0$ 。

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.3.4

将 $2 x$ 和 $0$ 相加。

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.4

对 $x$ 进行 $1$ 次方运算。

$- \frac{1}{5} \cdot \frac{x^{1} x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.5

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{1}{5} \cdot \frac{x^{1 + \frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.6

化简表达式。

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解题步骤 2.6.1

将 $1$ 写成具有公分母的分数。

$- \frac{1}{5} \cdot \frac{x^{\frac{5}{5} + \frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.6.2

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{x^{\frac{5 + 4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.6.3

将 $5$ 和 $4$ 相加。

$- \frac{1}{5} \cdot \frac{x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.6.4

将 $2$ 移到 $x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 的左侧。

$- \frac{1}{5} \cdot \frac{2 \cdot (x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.7

使用乘积法则求微分，根据该法则， $\frac{d}{d x} [f (x) g (x)]$ 等于 $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ ，其中 $f (x) = x^{\frac{4}{5}}$ 且 $g (x) = {(x^{2} - 1)}^{\frac{6}{5}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} \frac{d}{d x} [{(x^{2} - 1)}^{\frac{6}{5}}] + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.8

使用链式法则求微分，根据该法则， $\frac{d}{d x} [f (g (x))]$ 等于 $f' (g (x)) g' (x)$ ，其中 $f (x) = x^{\frac{6}{5}}$ 且 $g (x) = x^{2} - 1$ 。

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解题步骤 2.8.1

要使用链式法则，请将 $u$ 设为 $x^{2} - 1$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{d}{d u} [u^{\frac{6}{5}}] \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.8.2

使用幂法则求微分，根据该法则， $\frac{d}{d u} [u^{n}]$ 等于 $n u^{n - 1}$ ，其中 $n = \frac{6}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} u^{\frac{6}{5} - 1} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.8.3

使用 $x^{2} - 1$ 替换所有出现的 $u$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} - 1} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.9

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{5}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.10

组合 $- 1$ 和 $\frac{5}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.11

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6 - 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.12

化简分子。

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解题步骤 2.12.1

将 $- 1$ 乘以 $5$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6 - 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.12.2

从 $6$ 中减去 $5$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{1}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.13

合并分数。

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解题步骤 2.13.1

组合 $\frac{6}{5}$ 和 ${(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}}}{5} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.13.2

组合 $\frac{6 {(x^{2} - 1)}^{\frac{1}{5}}}{5}$ 和 $x^{\frac{4}{5}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} \frac{d}{d x} [x^{2} - 1] + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.14

根据加法法则， $x^{2} - 1$ 对 $x$ 的导数是 $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.15

使用幂法则求微分，根据该法则， $\frac{d}{d x} [x^{n}]$ 等于 $n x^{n - 1}$ ，其中 $n = 2$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x + \frac{d}{d x} [- 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.16

因为 $- 1$ 对于 $x$ 是常数，所以 $- 1$ 对 $x$ 的导数为 $0$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x + 0) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.17

合并分数。

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解题步骤 2.17.1

将 $2 x$ 和 $0$ 相加。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.17.2

组合 $2$ 和 $\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{2 (6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}})}{5} x + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.17.3

将 $6$ 乘以 $2$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}})}{5} x + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.17.4

组合 $\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}})}{5}$ 和 $x$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}) x}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.18

对 $x$ 进行 $1$ 次方运算。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} (x^{1} x^{\frac{4}{5}}))}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.19

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{1 + \frac{4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.20

化简表达式。

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解题步骤 2.20.1

将 $1$ 写成具有公分母的分数。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{5}{5} + \frac{4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.20.2

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{5 + 4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.20.3

将 $5$ 和 $4$ 相加。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.21

使用幂法则求微分，根据该法则， $\frac{d}{d x} [x^{n}]$ 等于 $n x^{n - 1}$ ，其中 $n = \frac{4}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.22

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{5}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.23

组合 $- 1$ 和 $\frac{5}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.24

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.25

化简分子。

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解题步骤 2.25.1

将 $- 1$ 乘以 $5$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4 - 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.25.2

从 $4$ 中减去 $5$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{- 1}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.26

将负号移到分数的前面。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{- \frac{1}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.27

组合 $\frac{4}{5}$ 和 $x^{- \frac{1}{5}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{4 x^{- \frac{1}{5}}}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.28

组合 ${(x^{2} - 1)}^{\frac{6}{5}}$ 和 $\frac{4 x^{- \frac{1}{5}}}{5}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{{(x^{2} - 1)}^{\frac{6}{5}} (4 x^{- \frac{1}{5}})}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.29

化简表达式。

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解题步骤 2.29.1

将 $4$ 移到 ${(x^{2} - 1)}^{\frac{6}{5}}$ 的左侧。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{4 \cdot {(x^{2} - 1)}^{\frac{6}{5}} x^{- \frac{1}{5}}}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.29.2

使用负指数规则 $b^{- n} = \frac{1}{b^{n}}$ 将 $x^{- \frac{1}{5}}$ 移动到分母。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.30

要将 $\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5}$ 写成带有公分母的分数，请乘以 $\frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} \cdot \frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.31

将 $\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5}$ 乘以 $\frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}} x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.32

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}} x^{\frac{1}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.33

通过指数相加将 $x^{\frac{9}{5}}$ 乘以 $x^{\frac{1}{5}}$ 。

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解题步骤 2.33.1

移动 $x^{\frac{1}{5}}$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} (x^{\frac{1}{5}} x^{\frac{9}{5}}) + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.33.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{1}{5} + \frac{9}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.33.3

在公分母上合并分子。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{1 + 9}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.33.4

将 $1$ 和 $9$ 相加。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{10}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.33.5

用 $10$ 除以 $5$ 。

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.34

将 $\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$ 乘以 $\frac{1}{5}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2} \cdot 5}$

解题步骤 2.35

将 $5$ 移到 ${(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}$ 的左侧。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 \cdot {(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

解题步骤 2.36

化简。

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解题步骤 2.36.1

对 $x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 运用乘积法则。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.2

运用分配律。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3

化简分子。

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解题步骤 2.36.3.1

化简分子。

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解题步骤 2.36.3.1.1

从 $12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}$ 中分解出因数 $4 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

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解题步骤 2.36.3.1.1.1

移动 ${(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{12 x^{2} {(x^{2} - 1)}^{\frac{1}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.1.2

从 $12 x^{2} {(x^{2} - 1)}^{\frac{1}{5}}$ 中分解出因数 $4 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.1.3

从 $4 {(x^{2} - 1)}^{\frac{6}{5}}$ 中分解出因数 $4 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{1}{5}} ({(x^{2} - 1)}^{\frac{5}{5}})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.1.4

从 $4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{1}{5}} ({(x^{2} - 1)}^{\frac{5}{5}})$ 中分解出因数 $4 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + {(x^{2} - 1)}^{\frac{5}{5}})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.2

用 $5$ 除以 $5$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + {(x^{2} - 1)}^{1})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.3

化简。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + x^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.4

将 $3 x^{2}$ 和 $x^{2}$ 相加。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (4 x^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.5

以因式分解的形式重写 $4 x^{2} - 1$ 。

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解题步骤 2.36.3.1.5.1

将 $4 x^{2}$ 重写为 ${(2 x)}^{2}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ({(2 x)}^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.5.2

将 $1$ 重写为 $1^{2}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ({(2 x)}^{2} - 1^{2})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.1.5.3

因为两项都是完全平方数，所以使用平方差公式 $a^{2} - b^{2} = (a + b) (a - b)$ 进行因式分解，其中 $a = 2 x$ 和 $b = 1$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ((2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.2

将 $- 1$ 乘以 $1$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.3

将 $- x^{2} - 1$ 乘以 $\frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + \frac{(- x^{2} - 1) (4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.4

将 $4$ 移到 $- x^{2} - 1$ 的左侧。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + \frac{4 \cdot (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.5

要将 $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 写成带有公分母的分数，请乘以 $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ 。

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot \frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}} + \frac{4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.6

组合 $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 和 $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ 。

$- \frac{\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}})}{5 x^{\frac{1}{5}}} + \frac{4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.7

在公分母上合并分子。

$- \frac{\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8

以因式分解的形式重写 $\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}$ 。

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解题步骤 2.36.3.8.1

从 $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)$ 中分解出因数 $2 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

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解题步骤 2.36.3.8.1.1

将表达式重新排序。

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解题步骤 2.36.3.8.1.1.1

移动 ${(x^{2} - 1)}^{\frac{6}{5}}$ 。

$- \frac{\frac{2 x^{\frac{9}{5}} \cdot 5 x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.1.1.2

移动 $x^{\frac{9}{5}}$ 。

$- \frac{\frac{2 \cdot 5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.1.2

从 $2 \cdot 5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ 中分解出因数 $2 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.1.3

从 $4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)$ 中分解出因数 $2 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 2 {(x^{2} - 1)}^{\frac{1}{5}} ((2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.1.4

从 $2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 2 {(x^{2} - 1)}^{\frac{1}{5}} ((2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))$ 中分解出因数 $2 {(x^{2} - 1)}^{\frac{1}{5}}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.2

通过指数相加将 $x^{\frac{9}{5}}$ 乘以 $x^{\frac{1}{5}}$ 。

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解题步骤 2.36.3.8.2.1

移动 $x^{\frac{1}{5}}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 (x^{\frac{1}{5}} x^{\frac{9}{5}}) {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{1}{5} + \frac{9}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.2.3

在公分母上合并分子。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{1 + 9}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.2.4

将 $1$ 和 $9$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{10}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.8.2.5

用 $10$ 除以 $5$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.9

约去 $5$ 的公因数。

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解题步骤 2.36.3.9.1

约去公因数。

解题步骤 2.36.3.9.2

重写表达式。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2} {(x^{2} - 1)}^{1} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10

化简分子。

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解题步骤 2.36.3.10.1

化简。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2} (x^{2} - 1) + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.2

运用分配律。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2} x^{2} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.3

通过指数相加将 $x^{2}$ 乘以 $x^{2}$ 。

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解题步骤 2.36.3.10.3.1

移动 $x^{2}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 (x^{2} x^{2}) + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.3.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2 + 2} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.3.3

将 $2$ 和 $2$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.4

将 $- 1$ 乘以 $5$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.5

运用分配律。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (2 (- x^{2}) + 2 \cdot - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.6

将 $- 1$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} + 2 \cdot - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.7

将 $2$ 乘以 $- 1$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} - 2) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.8

使用 FOIL 方法展开 $(- 2 x^{2} - 2) (2 x + 1)$ 。

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解题步骤 2.36.3.10.8.1

运用分配律。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} (2 x + 1) - 2 (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.8.2

运用分配律。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} (2 x) - 2 x^{2} \cdot 1 - 2 (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.8.3

运用分配律。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} (2 x) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9

化简每一项。

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解题步骤 2.36.3.10.9.1

使用乘法的交换性质重写。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 x^{2} x - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.2

通过指数相加将 $x^{2}$ 乘以 $x$ 。

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解题步骤 2.36.3.10.9.2.1

移动 $x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 (x \cdot x^{2}) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.2.2

将 $x$ 乘以 $x^{2}$ 。

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解题步骤 2.36.3.10.9.2.2.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 (x^{1} x^{2}) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.2.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 x^{1 + 2} - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.2.3

将 $1$ 和 $2$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 x^{3} - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.3

将 $- 2$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 4 x^{3} - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.4

将 $- 2$ 乘以 $1$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 4 x^{3} - 2 x^{2} - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.5

将 $2$ 乘以 $- 2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 4 x^{3} - 2 x^{2} - 4 x - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.9.6

将 $- 2$ 乘以 $1$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 4 x^{3} - 2 x^{2} - 4 x - 2) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.10

将第一个表达式中的每一项与第二个表达式中的每一项相乘来展开 $(- 4 x^{3} - 2 x^{2} - 4 x - 2) (2 x - 1)$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 x^{3} (2 x) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11

化简每一项。

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解题步骤 2.36.3.10.11.1

使用乘法的交换性质重写。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 x^{3} x - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.2

通过指数相加将 $x^{3}$ 乘以 $x$ 。

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解题步骤 2.36.3.10.11.2.1

移动 $x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 (x \cdot x^{3}) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.2.2

将 $x$ 乘以 $x^{3}$ 。

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解题步骤 2.36.3.10.11.2.2.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 (x^{1} x^{3}) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.2.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 x^{1 + 3} - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.2.3

将 $1$ 和 $3$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 x^{4} - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.3

将 $- 4$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.4

将 $- 1$ 乘以 $- 4$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.5

使用乘法的交换性质重写。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 \cdot 2 x^{2} x - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.6

通过指数相加将 $x^{2}$ 乘以 $x$ 。

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解题步骤 2.36.3.10.11.6.1

移动 $x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 \cdot 2 (x \cdot x^{2}) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.6.2

将 $x$ 乘以 $x^{2}$ 。

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解题步骤 2.36.3.10.11.6.2.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 \cdot 2 (x^{1} x^{2}) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.6.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 \cdot 2 x^{1 + 2} - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.6.3

将 $1$ 和 $2$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 2 \cdot 2 x^{3} - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.7

将 $- 2$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.8

将 $- 1$ 乘以 $- 2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.9

使用乘法的交换性质重写。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 4 \cdot 2 x \cdot x - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.10

通过指数相加将 $x$ 乘以 $x$ 。

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解题步骤 2.36.3.10.11.10.1

移动 $x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 4 \cdot 2 (x \cdot x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.10.2

将 $x$ 乘以 $x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 4 \cdot 2 x^{2} - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.11

将 $- 4$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.12

将 $- 1$ 乘以 $- 4$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} + 4 x - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.13

将 $2$ 乘以 $- 2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} + 4 x - 4 x - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.11.14

将 $- 2$ 乘以 $- 1$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.12

合并 $- 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2$ 中相反的项。

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解题步骤 2.36.3.10.12.1

从 $4 x^{3}$ 中减去 $4 x^{3}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 0 + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.12.2

将 $- 8 x^{4}$ 和 $0$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.12.3

从 $4 x$ 中减去 $4 x$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 2 x^{2} - 8 x^{2} + 0 + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.12.4

将 $- 8 x^{4} + 2 x^{2} - 8 x^{2}$ 和 $0$ 相加。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 2 x^{2} - 8 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.13

从 $2 x^{2}$ 中减去 $8 x^{2}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} - 6 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.14

从 $5 x^{4}$ 中减去 $8 x^{4}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 5 x^{2} - 6 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.3.10.15

从 $- 5 x^{2}$ 中减去 $6 x^{2}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.4

合并项。

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解题步骤 2.36.4.1

将 ${(x^{\frac{4}{5}})}^{2}$ 中的指数相乘。

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解题步骤 2.36.4.1.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{4}{5} \cdot 2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.4.1.2

乘以 $\frac{4}{5} \cdot 2$ 。

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解题步骤 2.36.4.1.2.1

组合 $\frac{4}{5}$ 和 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{4 \cdot 2}{5}} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.4.1.2.2

将 $4$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

解题步骤 2.36.4.2

将 ${({(x^{2} - 1)}^{\frac{6}{5}})}^{2}$ 中的指数相乘。

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解题步骤 2.36.4.2.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{6}{5} \cdot 2})}$

解题步骤 2.36.4.2.2

乘以 $\frac{6}{5} \cdot 2$ 。

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解题步骤 2.36.4.2.2.1

组合 $\frac{6}{5}$ 和 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{6 \cdot 2}{5}})}$

解题步骤 2.36.4.2.2.2

将 $6$ 乘以 $2$ 。

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

解题步骤 2.36.4.3

将 $\frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$ 重写为乘积形式。

$- (\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}} \cdot \frac{1}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}})$

解题步骤 2.36.4.4

将 $\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}$ 乘以 $\frac{1}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$ 。

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}} (5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

解题步骤 2.36.4.5

将 $5$ 乘以 $5$ 。

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{1}{5}} (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

解题步骤 2.36.4.6

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{8}{5} + \frac{1}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

解题步骤 2.36.4.7

在公分母上合并分子。

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{8 + 1}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

解题步骤 2.36.4.8

将 $8$ 和 $1$ 相加。

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

解题步骤 2.36.4.9

使用负指数规则 $b^{n} = \frac{1}{b^{- n}}$ 将 ${(x^{2} - 1)}^{\frac{1}{5}}$ 移动到分母。

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{12}{5}} {(x^{2} - 1)}^{- \frac{1}{5}}}$

解题步骤 2.36.4.10

通过指数相加将 ${(x^{2} - 1)}^{\frac{12}{5}}$ 乘以 ${(x^{2} - 1)}^{- \frac{1}{5}}$ 。

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解题步骤 2.36.4.10.1

移动 ${(x^{2} - 1)}^{- \frac{1}{5}}$ 。

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} ({(x^{2} - 1)}^{- \frac{1}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

解题步骤 2.36.4.10.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{- \frac{1}{5} + \frac{12}{5}}}$

解题步骤 2.36.4.10.3

在公分母上合并分子。

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{- 1 + 12}{5}}}$

解题步骤 2.36.4.10.4

将 $- 1$ 和 $12$ 相加。

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.5

从 $- 3 x^{4}$ 中分解出因数 $- 1$ 。

$- \frac{2 (- (3 x^{4}) - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.6

从 $- 11 x^{2}$ 中分解出因数 $- 1$ 。

$- \frac{2 (- (3 x^{4}) - (11 x^{2}) + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.7

从 $- (3 x^{4}) - (11 x^{2})$ 中分解出因数 $- 1$ 。

$- \frac{2 (- (3 x^{4} + 11 x^{2}) + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.8

将 $2$ 重写为 $- 1 (- 2)$ 。

$- \frac{2 (- (3 x^{4} + 11 x^{2}) - 1 (- 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.9

从 $- (3 x^{4} + 11 x^{2}) - 1 (- 2)$ 中分解出因数 $- 1$ 。

$- \frac{2 (- (3 x^{4} + 11 x^{2} - 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.10

将 $- (3 x^{4} + 11 x^{2} - 2)$ 重写为 $- 1 (3 x^{4} + 11 x^{2} - 2)$ 。

$- \frac{2 (- 1 (3 x^{4} + 11 x^{2} - 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.11

将负号移到分数的前面。

$- - \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.12

将 $- 1$ 乘以 $- 1$ 。

$1 \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

解题步骤 2.36.13

将 $\frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$ 乘以 $1$ 。

$f'' (x) = \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

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