微积分学示例

$\arccos (x)$

解题步骤 1

$\arccos (x)$ 对 $x$ 的导数为 $- \frac{1}{\sqrt{1 - x^{2}}}$ 。

$f' (x) = - \frac{1}{\sqrt{1 - x^{2}}}$

解题步骤 2

求二阶导数。

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解题步骤 2.1

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

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

解题步骤 2.2

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 $f (x) = - 1$ 且 $g (x) = \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}}$ 。

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

解题步骤 2.3

应用指数的基本规则。

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解题步骤 2.3.1

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

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

解题步骤 2.3.2

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

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解题步骤 2.3.2.1

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

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

解题步骤 2.3.2.2

组合 $\frac{1}{2}$ 和 $- 1$ 。

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

解题步骤 2.3.2.3

将负号移到分数的前面。

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

解题步骤 2.4

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 $f (x) = x^{- \frac{1}{2}}$ 且 $g (x) = 1 - x^{2}$ 。

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解题步骤 2.4.1

要使用链式法则，请将 $u$ 设为 $1 - x^{2}$ 。

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

解题步骤 2.4.2

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 $n = - \frac{1}{2}$ 。

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

解题步骤 2.4.3

使用 $1 - x^{2}$ 替换所有出现的 $u$ 。

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

解题步骤 2.5

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{2}{2}$ 。

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

解题步骤 2.6

组合 $- 1$ 和 $\frac{2}{2}$ 。

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

解题步骤 2.7

在公分母上合并分子。

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

解题步骤 2.8

化简分子。

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解题步骤 2.8.1

将 $- 1$ 乘以 $2$ 。

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

解题步骤 2.8.2

从 $- 1$ 中减去 $2$ 。

$- (- \frac{1}{2} {(1 - x^{2})}^{\frac{- 3}{2}} \frac{d}{d x} [1 - x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.9

合并分数。

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解题步骤 2.9.1

将负号移到分数的前面。

$- (- \frac{1}{2} {(1 - x^{2})}^{- \frac{3}{2}} \frac{d}{d x} [1 - x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.9.2

组合 ${(1 - x^{2})}^{- \frac{3}{2}}$ 和 $\frac{1}{2}$ 。

$- (- \frac{{(1 - x^{2})}^{- \frac{3}{2}}}{2} \frac{d}{d x} [1 - x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.9.3

化简表达式。

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解题步骤 2.9.3.1

使用负指数规则 $b^{- n} = \frac{1}{b^{n}}$ 将 ${(1 - x^{2})}^{- \frac{3}{2}}$ 移动到分母。

$- (- \frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [1 - x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.9.3.2

将 $- 1$ 乘以 $- 1$ 。

$1 (\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [1 - x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.9.3.3

将 $\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}}$ 乘以 $1$ 。

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [1 - x^{2}] + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.10

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

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.11

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

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.12

将 $0$ 和 $\frac{d}{d x} [- x^{2}]$ 相加。

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- x^{2}] + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.13

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

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} (- \frac{d}{d x} [x^{2}]) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.14

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 $n = 2$ 。

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} (- (2 x)) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.15

化简项。

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解题步骤 2.15.1

将 $2$ 乘以 $- 1$ 。

$\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}} (- 2 x) + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.15.2

组合 $- 2$ 和 $\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}}$ 。

$\frac{- 2}{2 {(1 - x^{2})}^{\frac{3}{2}}} x + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.15.3

组合 $\frac{- 2}{2 {(1 - x^{2})}^{\frac{3}{2}}}$ 和 $x$ 。

$\frac{- 2 x}{2 {(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.15.4

从 $- 2 x$ 中分解出因数 $2$ 。

$\frac{2 (- x)}{2 {(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.16

约去公因数。

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解题步骤 2.16.1

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

$\frac{2 (- x)}{2 ({(1 - x^{2})}^{\frac{3}{2}})} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.16.2

约去公因数。

$\frac{2 (- x)}{2 {(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.16.3

重写表达式。

$\frac{- x}{{(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.17

将负号移到分数的前面。

$- \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

解题步骤 2.18

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

$- \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} + \frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} \cdot 0$

解题步骤 2.19

化简表达式。

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解题步骤 2.19.1

将 $\frac{1}{{(1 - x^{2})}^{\frac{1}{2}}}$ 乘以 $0$ 。

$- \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} + 0$

解题步骤 2.19.2

将 $- \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ 和 $0$ 相加。

$f'' (x) = - \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$

解题步骤 3

求三阶导数。

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解题步骤 3.1

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 $f (x) = - 1$ 且 $g (x) = \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ 。

$- \frac{d}{d x} [\frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}] + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.2

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 $f (x) = x$ 且 $g (x) = {(1 - x^{2})}^{\frac{3}{2}}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{({(1 - x^{2})}^{\frac{3}{2}})}^{2}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.3

使用幂法则求微分。

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解题步骤 3.3.1

将 ${({(1 - x^{2})}^{\frac{3}{2}})}^{2}$ 中的指数相乘。

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解题步骤 3.3.1.1

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{(1 - x^{2})}^{\frac{3}{2} \cdot 2}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.3.1.2

约去 $2$ 的公因数。

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解题步骤 3.3.1.2.1

约去公因数。

解题步骤 3.3.1.2.2

重写表达式。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.3.2

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 $n = 1$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} \cdot 1 - x \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.3.3

将 ${(1 - x^{2})}^{\frac{3}{2}}$ 乘以 $1$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.4

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 $f (x) = x^{\frac{3}{2}}$ 且 $g (x) = 1 - x^{2}$ 。

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解题步骤 3.4.1

要使用链式法则，请将 $u$ 设为 $1 - x^{2}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.4.2

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 $n = \frac{3}{2}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.4.3

使用 $1 - x^{2}$ 替换所有出现的 $u$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} - 1} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.5

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.6

组合 $- 1$ 和 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.7

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.8

化简分子。

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解题步骤 3.8.1

将 $- 1$ 乘以 $2$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{3 - 2}{2}} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.8.2

从 $3$ 中减去 $2$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3}{2} {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.9

合并分数。

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解题步骤 3.9.1

组合 $\frac{3}{2}$ 和 ${(1 - x^{2})}^{\frac{1}{2}}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - x (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.9.2

组合 $\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2}$ 和 $x$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.10

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.11

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} (0 + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.12

将 $0$ 和 $\frac{d}{d x} [- x^{2}]$ 相加。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} \frac{d}{d x} [- x^{2}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.13

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} - \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} (- \frac{d}{d x} [x^{2}])}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.14

乘。

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解题步骤 3.14.1

将 $- 1$ 乘以 $- 1$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 1 \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.14.2

将 $\frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2}$ 乘以 $1$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.15

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 $n = 2$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2} (2 x)}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.16

合并分数。

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解题步骤 3.16.1

组合 $2$ 和 $\frac{3 {(1 - x^{2})}^{\frac{1}{2}} x}{2}$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{2 (3 {(1 - x^{2})}^{\frac{1}{2}} x)}{2} x}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.16.2

将 $3$ 乘以 $2$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} x)}{2} x}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.16.3

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} x) x}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.17

对 $x$ 进行 $1$ 次方运算。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} (x^{1} x))}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.18

对 $x$ 进行 $1$ 次方运算。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} (x^{1} x^{1}))}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.19

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} x^{1 + 1})}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.20

将 $1$ 和 $1$ 相加。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{6 ({(1 - x^{2})}^{\frac{1}{2}} x^{2})}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.21

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{2 (3 {(1 - x^{2})}^{\frac{1}{2}} x^{2})}{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.22

约去公因数。

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解题步骤 3.22.1

从 $2$ 中分解出因数 $2$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{2 (3 {(1 - x^{2})}^{\frac{1}{2}} x^{2})}{2 (1)}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.22.2

约去公因数。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{2 (3 {(1 - x^{2})}^{\frac{1}{2}} x^{2})}{2 \cdot 1}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.22.3

重写表达式。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + \frac{3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{1}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.22.4

用 $3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ 除以 $1$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

解题步骤 3.23

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

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} + \frac{x}{{(1 - x^{2})}^{\frac{3}{2}}} \cdot 0$

解题步骤 3.24

化简表达式。

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解题步骤 3.24.1

将 $\frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ 乘以 $0$ 。

$- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} + 0$

解题步骤 3.24.2

将 $- \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$ 和 $0$ 相加。

$f''' (x) = - \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$

解题步骤 4

求四阶导数。

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解题步骤 4.1

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 $f (x) = - 1$ 且 $g (x) = \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$ 。

$- \frac{d}{d x} [\frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}] + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.2

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 $f (x) = {(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ 且 $g (x) = {(1 - x^{2})}^{3}$ 。

$- \frac{{(1 - x^{2})}^{3} \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}] - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{({(1 - x^{2})}^{3})}^{2}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.3

使用求加法法则求导。

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解题步骤 4.3.1

将 ${({(1 - x^{2})}^{3})}^{2}$ 中的指数相乘。

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解题步骤 4.3.1.1

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

$- \frac{{(1 - x^{2})}^{3} \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}] - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{3 \cdot 2}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.3.1.2

将 $3$ 乘以 $2$ 。

$- \frac{{(1 - x^{2})}^{3} \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}] - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.3.2

根据加法法则， ${(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ 对 $x$ 的导数是 $\frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.4

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 $f (x) = x^{\frac{3}{2}}$ 且 $g (x) = 1 - x^{2}$ 。

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解题步骤 4.4.1

要使用链式法则，请将 $u_{1}$ 设为 $1 - x^{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{d}{d u_{1}} [{u_{1}}^{\frac{3}{2}}] \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.4.2

使用幂法则求微分，根据该法则， $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ 等于 $n {u_{1}}^{n - 1}$ ，其中 $n = \frac{3}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {u_{1}}^{\frac{3}{2} - 1} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.4.3

使用 $1 - x^{2}$ 替换所有出现的 $u_{1}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} - 1} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.5

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.6

组合 $- 1$ 和 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.7

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.8

化简分子。

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解题步骤 4.8.1

将 $- 1$ 乘以 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{3 - 2}{2}} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.8.2

从 $3$ 中减去 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3}{2} {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.9

组合 $\frac{3}{2}$ 和 ${(1 - x^{2})}^{\frac{1}{2}}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} \frac{d}{d x} [1 - x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.10

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

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]) + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.11

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

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (0 + \frac{d}{d x} [- x^{2}]) + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.12

将 $0$ 和 $\frac{d}{d x} [- x^{2}]$ 相加。

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.13

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

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (- \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.14

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 $n = 2$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (- (2 x)) + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.15

化简项。

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解题步骤 4.15.1

将 $2$ 乘以 $- 1$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (- 2 x) + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.15.2

组合 $- 2$ 和 $\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{- 2 (3 {(1 - x^{2})}^{\frac{1}{2}})}{2} x + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.15.3

将 $3$ 乘以 $- 2$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{- 6 {(1 - x^{2})}^{\frac{1}{2}}}{2} x + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.15.4

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

$- \frac{{(1 - x^{2})}^{3} (\frac{- 6 {(1 - x^{2})}^{\frac{1}{2}} x}{2} + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.15.5

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

$- \frac{{(1 - x^{2})}^{3} (\frac{2 (- 3 {(1 - x^{2})}^{\frac{1}{2}} x)}{2} + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.16

约去公因数。

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解题步骤 4.16.1

从 $2$ 中分解出因数 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{2 (- 3 {(1 - x^{2})}^{\frac{1}{2}} x)}{2 (1)} + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.16.2

约去公因数。

$- \frac{{(1 - x^{2})}^{3} (\frac{2 (- 3 {(1 - x^{2})}^{\frac{1}{2}} x)}{2 \cdot 1} + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.16.3

重写表达式。

$- \frac{{(1 - x^{2})}^{3} (\frac{- 3 {(1 - x^{2})}^{\frac{1}{2}} x}{1} + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.16.4

用 $- 3 {(1 - x^{2})}^{\frac{1}{2}} x$ 除以 $1$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{d}{d x} [3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.17

因为 $3$ 对于 $x$ 是常数，所以 $3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}$ 对 $x$ 的导数是 $3 \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}} x^{2}]$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}} x^{2}]) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.18

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 $f (x) = {(1 - x^{2})}^{\frac{1}{2}}$ 且 $g (x) = x^{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 ({(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}}])) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.19

使用幂法则求微分。

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解题步骤 4.19.1

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 $n = 2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 ({(1 - x^{2})}^{\frac{1}{2}} (2 x) + x^{2} \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}}])) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.19.2

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 \cdot {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} \frac{d}{d x} [{(1 - x^{2})}^{\frac{1}{2}}])) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.20

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 $f (x) = x^{\frac{1}{2}}$ 且 $g (x) = 1 - x^{2}$ 。

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解题步骤 4.20.1

要使用链式法则，请将 $u_{2}$ 设为 $1 - x^{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.20.2

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.20.3

使用 $1 - x^{2}$ 替换所有出现的 $u_{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.21

要将 $- 1$ 写成带有公分母的分数，请乘以 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.22

组合 $- 1$ 和 $\frac{2}{2}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.23

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.24

化简分子。

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解题步骤 4.24.1

将 $- 1$ 乘以 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.24.2

从 $1$ 中减去 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.25

合并分数。

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解题步骤 4.25.1

将负号移到分数的前面。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2} {(1 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.25.2

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{{(1 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.25.3

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + x^{2} (\frac{1}{2 {(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 - x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.25.4

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 - x^{2}])) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.26

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.27

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.28

将 $0$ 和 $\frac{d}{d x} [- x^{2}]$ 相加。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}])) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.29

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.30

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 $n = 2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} (- (2 x)))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.31

合并分数。

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解题步骤 4.31.1

将 $2$ 乘以 $- 1$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} (- 2 x))) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.31.2

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- 2 x^{2}}{2 {(1 - x^{2})}^{\frac{1}{2}}} x)) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.31.3

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- 2 x^{2} x}{2 {(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.32

对 $x$ 进行 $1$ 次方运算。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- 2 (x^{1} x^{2})}{2 {(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.33

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- 2 x^{1 + 2}}{2 {(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.34

将 $1$ 和 $2$ 相加。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- 2 x^{3}}{2 {(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.35

从 $- 2 x^{3}$ 中分解出因数 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{2 (- x^{3})}{2 {(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.36

约去公因数。

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解题步骤 4.36.1

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{2 (- x^{3})}{2 ({(1 - x^{2})}^{\frac{1}{2}})})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.36.2

约去公因数。

解题步骤 4.36.3

重写表达式。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{- x^{3}}{{(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.37

将负号移到分数的前面。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 {(1 - x^{2})}^{\frac{1}{2}} x - \frac{x^{3}}{{(1 - x^{2})}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.38

将 $1$ 和 $- x^{2}$ 重新排序。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (2 x {(- x^{2} + 1)}^{\frac{1}{2}} - \frac{x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.39

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

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 (\frac{2 x {(- x^{2} + 1)}^{\frac{1}{2}} {(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}} - \frac{x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.40

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x {(- x^{2} + 1)}^{\frac{1}{2}} {(- x^{2} + 1)}^{\frac{1}{2}} - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.41

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

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解题步骤 4.41.1

移动 ${(- x^{2} + 1)}^{\frac{1}{2}}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x ({(- x^{2} + 1)}^{\frac{1}{2}} {(- x^{2} + 1)}^{\frac{1}{2}}) - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.41.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x {(- x^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}} - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.41.3

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x {(- x^{2} + 1)}^{\frac{1 + 1}{2}} - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.41.4

将 $1$ 和 $1$ 相加。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x {(- x^{2} + 1)}^{\frac{2}{2}} - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.41.5

用 $2$ 除以 $2$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x {(- x^{2} + 1)}^{1} - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.42

化简 $2 x {(- x^{2} + 1)}^{1}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + 3 \frac{2 x (- x^{2} + 1) - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.43

组合 $3$ 和 $\frac{2 x (- x^{2} + 1) - x^{3}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x + \frac{3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.44

要将 $- 3 {(1 - x^{2})}^{\frac{1}{2}} x$ 写成带有公分母的分数，请乘以 $\frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ 。

$- \frac{{(1 - x^{2})}^{3} (- 3 {(1 - x^{2})}^{\frac{1}{2}} x \cdot \frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}} + \frac{3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.45

组合 $- 3 {(1 - x^{2})}^{\frac{1}{2}} x$ 和 $\frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ 。

$- \frac{{(1 - x^{2})}^{3} (\frac{- 3 {(1 - x^{2})}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}} + \frac{3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}}) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.46

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(1 - x^{2})}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}} + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.47

重新排序项。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}} + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.48

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

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解题步骤 4.48.1

移动 ${(- x^{2} + 1)}^{\frac{1}{2}}$ 。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 ({(- x^{2} + 1)}^{\frac{1}{2}} {(- x^{2} + 1)}^{\frac{1}{2}}) x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.48.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(- x^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}} x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.48.3

在公分母上合并分子。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(- x^{2} + 1)}^{\frac{1 + 1}{2}} x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.48.4

将 $1$ 和 $1$ 相加。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(- x^{2} + 1)}^{\frac{2}{2}} x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.48.5

用 $2$ 除以 $2$ 。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 {(- x^{2} + 1)}^{1} x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.49

化简 $- 3 {(- x^{2} + 1)}^{1} x$ 。

$- \frac{{(1 - x^{2})}^{3} \frac{- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.50

组合 ${(1 - x^{2})}^{3}$ 和 $\frac{- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ 。

$- \frac{\frac{{(1 - x^{2})}^{3} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3}))}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.51

重新排序项。

$- \frac{\frac{{(- x^{2} + 1)}^{3} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3}))}{{(- x^{2} + 1)}^{\frac{1}{2}}} - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{3}]}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.52

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

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

解题步骤 4.53

约去公因数。

点击获取更多步骤...

解题步骤 4.53.1

乘以 $1$ 。

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

解题步骤 4.53.2

约去公因数。

解题步骤 4.53.3

重写表达式。

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

解题步骤 4.53.4

用 ${(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3}))$ 除以 $1$ 。

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

解题步骤 4.54

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 $f (x) = x^{3}$ 且 $g (x) = 1 - x^{2}$ 。

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解题步骤 4.54.1

要使用链式法则，请将 $u_{3}$ 设为 $1 - x^{2}$ 。

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (\frac{d}{d u_{3}} [{u_{3}}^{3}] \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.54.2

使用幂法则求微分，根据该法则， $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ 等于 $n {u_{3}}^{n - 1}$ ，其中 $n = 3$ 。

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (3 {u_{3}}^{2} \frac{d}{d x} [1 - x^{2}])}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.54.3

使用 $1 - x^{2}$ 替换所有出现的 $u_{3}$ 。

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

解题步骤 4.55

求微分。

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解题步骤 4.55.1

将 $3$ 乘以 $- 1$ 。

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

解题步骤 4.55.2

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

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

解题步骤 4.55.3

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

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) - 3 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} (0 + \frac{d}{d x} [- x^{2}]))}{{(1 - x^{2})}^{6}} + \frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}} \frac{d}{d x} [- 1]$

解题步骤 4.55.4

将 $0$ 和 $\frac{d}{d x} [- x^{2}]$ 相加。

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

解题步骤 4.55.5

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

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

解题步骤 4.55.6

将 $- 1$ 乘以 $- 3$ 。

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

解题步骤 4.55.7

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 $n = 2$ 。

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

解题步骤 4.55.8

化简表达式。

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解题步骤 4.55.8.1

将 $2$ 移到 ${(1 - x^{2})}^{2}$ 的左侧。

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

解题步骤 4.55.8.2

将 $2$ 乘以 $3$ 。

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

解题步骤 4.55.9

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

解题步骤 4.55.10

化简表达式。

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解题步骤 4.55.10.1

将 $\frac{{(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}}{{(1 - x^{2})}^{3}}$ 乘以 $0$ 。

解题步骤 4.55.10.2

将 $- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 (- x^{2} + 1) x + 3 (2 x (- x^{2} + 1) - x^{3})) + 6 ({(1 - x^{2})}^{\frac{3}{2}} + 3 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}}$ 和 $0$ 相加。

解题步骤 4.56

化简。

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解题步骤 4.56.1

运用分配律。

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

解题步骤 4.56.2

运用分配律。

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

解题步骤 4.56.3

运用分配律。

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

解题步骤 4.56.4

运用分配律。

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

解题步骤 4.56.5

运用分配律。

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

解题步骤 4.56.6

化简分子。

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解题步骤 4.56.6.1

化简每一项。

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解题步骤 4.56.6.1.1

通过指数相加将 $x^{2}$ 乘以 $x$ 。

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解题步骤 4.56.6.1.1.1

移动 $x$ 。

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

解题步骤 4.56.6.1.1.2

将 $x$ 乘以 $x^{2}$ 。

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解题步骤 4.56.6.1.1.2.1

对 $x$ 进行 $1$ 次方运算。

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

解题步骤 4.56.6.1.1.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 4.56.6.1.1.3

将 $1$ 和 $2$ 相加。

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

解题步骤 4.56.6.1.2

将 $- 1$ 乘以 $- 3$ 。

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

解题步骤 4.56.6.1.3

将 $- 3$ 乘以 $1$ 。

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

解题步骤 4.56.6.1.4

通过指数相加将 $x$ 乘以 $x^{2}$ 。

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解题步骤 4.56.6.1.4.1

移动 $x^{2}$ 。

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

解题步骤 4.56.6.1.4.2

将 $x^{2}$ 乘以 $x$ 。

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解题步骤 4.56.6.1.4.2.1

对 $x$ 进行 $1$ 次方运算。

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

解题步骤 4.56.6.1.4.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 4.56.6.1.4.3

将 $2$ 和 $1$ 相加。

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

解题步骤 4.56.6.1.5

将 $- 1$ 乘以 $2$ 。

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

解题步骤 4.56.6.1.6

将 $- 2$ 乘以 $3$ 。

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

解题步骤 4.56.6.1.7

将 $2$ 乘以 $1$ 。

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

解题步骤 4.56.6.1.8

将 $2$ 乘以 $3$ 。

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

解题步骤 4.56.6.1.9

将 $- 1$ 乘以 $3$ 。

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

解题步骤 4.56.6.2

合并 $3 x^{3} - 3 x - 6 x^{3} + 6 x - 3 x^{3}$ 中相反的项。

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解题步骤 4.56.6.2.1

从 $3 x^{3}$ 中减去 $3 x^{3}$ 。

$- \frac{{(- x^{2} + 1)}^{\frac{5}{2}} (- 3 x - 6 x^{3} + 6 x + 0) + (6 {(1 - x^{2})}^{\frac{3}{2}} + 6 (3 {(1 - x^{2})}^{\frac{1}{2}} x^{2})) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.2.2

将 $- 3 x - 6 x^{3} + 6 x$ 和 $0$ 相加。

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

解题步骤 4.56.6.3

将 $- 3 x$ 和 $6 x$ 相加。

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

解题步骤 4.56.6.4

运用分配律。

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

解题步骤 4.56.6.5

使用乘法的交换性质重写。

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

解题步骤 4.56.6.6

使用乘法的交换性质重写。

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

解题步骤 4.56.6.7

将 $3$ 乘以 $6$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ({(1 - x^{2})}^{2} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.8

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

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2}) (1 - x^{2}) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.9

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

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解题步骤 4.56.6.9.1

运用分配律。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 (1 - x^{2}) - x^{2} (1 - x^{2})) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.9.2

运用分配律。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 \cdot 1 + 1 (- x^{2}) - x^{2} (1 - x^{2})) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.9.3

运用分配律。

解题步骤 4.56.6.10

化简并合并同类项。

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解题步骤 4.56.6.10.1

化简每一项。

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解题步骤 4.56.6.10.1.1

将 $1$ 乘以 $1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 + 1 (- x^{2}) - x^{2} \cdot 1 - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.1.2

将 $- x^{2}$ 乘以 $1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} \cdot 1 - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.1.3

将 $- 1$ 乘以 $1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} - x^{2} (- x^{2})) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.1.4

使用乘法的交换性质重写。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} - 1 \cdot - 1 x^{2} x^{2}) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.1.5

通过指数相加将 $x^{2}$ 乘以 $x^{2}$ 。

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解题步骤 4.56.6.10.1.5.1

移动 $x^{2}$ 。

解题步骤 4.56.6.10.1.5.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

解题步骤 4.56.6.10.1.5.3

将 $2$ 和 $2$ 相加。

解题步骤 4.56.6.10.1.6

将 $- 1$ 乘以 $- 1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} + 1 x^{4}) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.1.7

将 $x^{4}$ 乘以 $1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - x^{2} - x^{2} + x^{4}) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.10.2

从 $- x^{2}$ 中减去 $x^{2}$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) ((1 - 2 x^{2} + x^{4}) x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.11

运用分配律。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (1 x - 2 x^{2} x + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12

化简。

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解题步骤 4.56.6.12.1

将 $x$ 乘以 $1$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{2} x + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.2

通过指数相加将 $x^{2}$ 乘以 $x$ 。

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解题步骤 4.56.6.12.2.1

移动 $x$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 (x \cdot x^{2}) + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.2.2

将 $x$ 乘以 $x^{2}$ 。

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解题步骤 4.56.6.12.2.2.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 (x^{1} x^{2}) + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.2.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{1 + 2} + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.2.3

将 $1$ 和 $2$ 相加。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4} x)}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.3

通过指数相加将 $x^{4}$ 乘以 $x$ 。

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解题步骤 4.56.6.12.3.1

将 $x^{4}$ 乘以 $x$ 。

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解题步骤 4.56.6.12.3.1.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4} x^{1})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.3.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{4 + 1})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.12.3.2

将 $4$ 和 $1$ 相加。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + (6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{5})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.13

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 $(6 {(1 - x^{2})}^{\frac{3}{2}} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) (x - 2 x^{3} + x^{5})$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} (- 2 x^{3}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14

化简每一项。

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解题步骤 4.56.6.14.1

使用乘法的交换性质重写。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x + 6 \cdot - 2 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.2

将 $6$ 乘以 $- 2$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.3

通过指数相加将 $x^{2}$ 乘以 $x$ 。

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解题步骤 4.56.6.14.3.1

移动 $x$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x \cdot x^{2}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.3.2

将 $x$ 乘以 $x^{2}$ 。

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解题步骤 4.56.6.14.3.2.1

对 $x$ 进行 $1$ 次方运算。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{1} x^{2}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.3.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{1 + 2} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.3.3

将 $1$ 和 $2$ 相加。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} (- 2 x^{3}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.4

通过指数相加将 $x^{2}$ 乘以 $x^{3}$ 。

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解题步骤 4.56.6.14.4.1

移动 $x^{3}$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{3} x^{2}) \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.4.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3 + 2} \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.4.3

将 $3$ 和 $2$ 相加。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{5} \cdot - 2 + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.5

将 $- 2$ 乘以 $18$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{2} x^{5}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.6

通过指数相加将 $x^{2}$ 乘以 $x^{5}$ 。

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解题步骤 4.56.6.14.6.1

移动 $x^{5}$ 。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} (x^{5} x^{2})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.6.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{5 + 2}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.14.6.3

将 $5$ 和 $2$ 相加。

$- \frac{- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15

以因式分解的形式重写 $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ 。

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解题步骤 4.56.6.15.1

从 $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3} + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ 中分解出因数 $3 x$ 。

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解题步骤 4.56.6.15.1.1

从 $- 6 {(- x^{2} + 1)}^{\frac{5}{2}} x^{3}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 {(- x^{2} + 1)}^{\frac{5}{2}} x + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.2

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

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.3

从 $6 {(1 - x^{2})}^{\frac{3}{2}} x$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) - 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3} + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.4

从 $- 12 {(1 - x^{2})}^{\frac{3}{2}} x^{3}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 6 {(1 - x^{2})}^{\frac{3}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.5

从 $6 {(1 - x^{2})}^{\frac{3}{2}} x^{5}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{3} - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.6

从 $18 {(1 - x^{2})}^{\frac{1}{2}} x^{3}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) - 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5} + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.7

从 $- 36 {(1 - x^{2})}^{\frac{1}{2}} x^{5}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.8

从 $18 {(1 - x^{2})}^{\frac{1}{2}} x^{7}$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2}) + 3 x ({(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.9

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

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.10

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

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}}) + 3 x (- 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.11

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

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2}) + 3 x (2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.12

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

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.13

从 $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{2})$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.14

从 $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{2}) + 3 x (- 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4})$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{2} - 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.1.15

从 $3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{2} - 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4}) + 3 x (6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})$ 中分解出因数 $3 x$ 。

$- \frac{3 x (- 2 {(- x^{2} + 1)}^{\frac{5}{2}} x^{2} + {(- x^{2} + 1)}^{\frac{5}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} - 4 {(1 - x^{2})}^{\frac{3}{2}} x^{2} + 2 {(1 - x^{2})}^{\frac{3}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{2} - 12 {(1 - x^{2})}^{\frac{1}{2}} x^{4} + 6 {(1 - x^{2})}^{\frac{1}{2}} x^{6})}{{(1 - x^{2})}^{6}}$

解题步骤 4.56.6.15.2

重新排序项。

$f^{4} (x) = - \frac{3 x (- 2 x^{2} {(- x^{2} + 1)}^{\frac{5}{2}} - 4 x^{2} {(1 - x^{2})}^{\frac{3}{2}} + 2 x^{4} {(1 - x^{2})}^{\frac{3}{2}} + 6 x^{2} {(1 - x^{2})}^{\frac{1}{2}} - 12 x^{4} {(1 - x^{2})}^{\frac{1}{2}} + 6 x^{6} {(1 - x^{2})}^{\frac{1}{2}} + 2 {(1 - x^{2})}^{\frac{3}{2}} + {(- x^{2} + 1)}^{\frac{5}{2}})}{{(1 - x^{2})}^{6}}$

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