Cálculo Exemplos

$y = \frac{\sqrt{a + x} - \sqrt{a - x}}{\sqrt{a + x} + \sqrt{a - x}}$

Etapa 1

Aplique regras básicas de expoentes.

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Etapa 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{a + x}$ como ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{a - x}$ como ${(a - x)}^{\frac{1}{2}}$ .

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

Etapa 1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{a + x}$ como ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{a - x}$ como ${(a - x)}^{\frac{1}{2}}$ .

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

Etapa 2

Diferencie usando a regra do quociente, que determina que $\frac{d}{d a} [\frac{f (a)}{g (a)}]$ é $\frac{g (a) \frac{d}{d a} [f (a)] - f (a) \frac{d}{d a} [g (a)]}{{g (a)}^{2}}$ , em que $f (a) = {(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ e $g (a) = {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ .

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

Etapa 3

De acordo com a regra da soma, a derivada de ${(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ com relação a $a$ é $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [- {(a - x)}^{\frac{1}{2}}]$ .

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

Etapa 4

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d a} [f (g (a))]$ é $f' (g (a)) g' (a)$ , em que $f (a) = a^{\frac{1}{2}}$ e $g (a) = a + x$ .

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Etapa 4.1

Para aplicar a regra da cadeia, defina $u_{1}$ como $a + x$ .

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

Etapa 4.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ é $n {u_{1}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

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

Etapa 4.3

Substitua todas as ocorrências de $u_{1}$ por $a + x$ .

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

Etapa 5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

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

Etapa 6

Combine $- 1$ e $\frac{2}{2}$ .

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

Etapa 7

Combine os numeradores em relação ao denominador comum.

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

Etapa 8

Simplifique o numerador.

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Etapa 8.1

Multiplique $- 1$ por $2$ .

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

Etapa 8.2

Subtraia $2$ de $1$ .

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

Etapa 9

Diferencie.

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Etapa 9.1

Mova o número negativo para a frente da fração.

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

Etapa 9.2

Combine frações.

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Etapa 9.2.1

Combine $\frac{1}{2}$ e ${(a + x)}^{- \frac{1}{2}}$ .

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

Etapa 9.2.2

Mova ${(a + x)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Etapa 9.3

De acordo com a regra da soma, a derivada de $a + x$ com relação a $a$ é $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Etapa 9.4

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d a} [a^{n}]$ é $n a^{n - 1}$ , em que $n = 1$ .

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

Etapa 9.5

Como $x$ é constante em relação a $a$ , a derivada de $x$ em relação a $a$ é $0$ .

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

Etapa 9.6

Simplifique a expressão.

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Etapa 9.6.1

Some $1$ e $0$ .

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

Etapa 9.6.2

Multiplique $\frac{1}{2 {(a + x)}^{\frac{1}{2}}}$ por $1$ .

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

Etapa 9.7

Como $- 1$ é constante em relação a $a$ , a derivada de $- {(a - x)}^{\frac{1}{2}}$ em relação a $a$ é $- \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Etapa 10

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d a} [f (g (a))]$ é $f' (g (a)) g' (a)$ , em que $f (a) = a^{\frac{1}{2}}$ e $g (a) = a - x$ .

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Etapa 10.1

Para aplicar a regra da cadeia, defina $u_{2}$ como $a - x$ .

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

Etapa 10.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ é $n {u_{2}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

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

Etapa 10.3

Substitua todas as ocorrências de $u_{2}$ por $a - x$ .

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

Etapa 11

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

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

Etapa 12

Combine $- 1$ e $\frac{2}{2}$ .

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

Etapa 13

Combine os numeradores em relação ao denominador comum.

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

Etapa 14

Simplifique o numerador.

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Etapa 14.1

Multiplique $- 1$ por $2$ .

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

Etapa 14.2

Subtraia $2$ de $1$ .

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

Etapa 15

Diferencie.

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Etapa 15.1

Mova o número negativo para a frente da fração.

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

Etapa 15.2

Combine frações.

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Etapa 15.2.1

Combine $\frac{1}{2}$ e ${(a - x)}^{- \frac{1}{2}}$ .

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

Etapa 15.2.2

Mova ${(a - x)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Etapa 15.3

De acordo com a regra da soma, a derivada de $a - x$ com relação a $a$ é $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Etapa 15.4

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d a} [a^{n}]$ é $n a^{n - 1}$ , em que $n = 1$ .

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

Etapa 15.5

Como $- x$ é constante em relação a $a$ , a derivada de $- x$ em relação a $a$ é $0$ .

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

Etapa 15.6

Simplifique a expressão.

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Etapa 15.6.1

Some $1$ e $0$ .

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

Etapa 15.6.2

Multiplique $- 1$ por $1$ .

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

Etapa 15.7

De acordo com a regra da soma, a derivada de ${(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ com relação a $a$ é $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Etapa 16

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d a} [f (g (a))]$ é $f' (g (a)) g' (a)$ , em que $f (a) = a^{\frac{1}{2}}$ e $g (a) = a + x$ .

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Etapa 16.1

Para aplicar a regra da cadeia, defina $u_{3}$ como $a + x$ .

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

Etapa 16.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ é $n {u_{3}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

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

Etapa 16.3

Substitua todas as ocorrências de $u_{3}$ por $a + x$ .

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

Etapa 17

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

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

Etapa 18

Combine $- 1$ e $\frac{2}{2}$ .

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

Etapa 19

Combine os numeradores em relação ao denominador comum.

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

Etapa 20

Simplifique o numerador.

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Etapa 20.1

Multiplique $- 1$ por $2$ .

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

Etapa 20.2

Subtraia $2$ de $1$ .

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

Etapa 21

Diferencie.

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Etapa 21.1

Mova o número negativo para a frente da fração.

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

Etapa 21.2

Combine frações.

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Etapa 21.2.1

Combine $\frac{1}{2}$ e ${(a + x)}^{- \frac{1}{2}}$ .

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

Etapa 21.2.2

Mova ${(a + x)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Etapa 21.3

De acordo com a regra da soma, a derivada de $a + x$ com relação a $a$ é $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Etapa 21.4

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d a} [a^{n}]$ é $n a^{n - 1}$ , em que $n = 1$ .

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

Etapa 21.5

Como $x$ é constante em relação a $a$ , a derivada de $x$ em relação a $a$ é $0$ .

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

Etapa 21.6

Simplifique a expressão.

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Etapa 21.6.1

Some $1$ e $0$ .

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

Etapa 21.6.2

Multiplique $\frac{1}{2 {(a + x)}^{\frac{1}{2}}}$ por $1$ .

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

Etapa 22

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d a} [f (g (a))]$ é $f' (g (a)) g' (a)$ , em que $f (a) = a^{\frac{1}{2}}$ e $g (a) = a - x$ .

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Etapa 22.1

Para aplicar a regra da cadeia, defina $u_{4}$ como $a - x$ .

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

Etapa 22.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ é $n {u_{4}}^{n - 1}$ , em que $n = \frac{1}{2}$ .

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

Etapa 22.3

Substitua todas as ocorrências de $u_{4}$ por $a - x$ .

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

Etapa 23

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

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

Etapa 24

Combine $- 1$ e $\frac{2}{2}$ .

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

Etapa 25

Combine os numeradores em relação ao denominador comum.

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

Etapa 26

Simplifique o numerador.

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Etapa 26.1

Multiplique $- 1$ por $2$ .

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

Etapa 26.2

Subtraia $2$ de $1$ .

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

Etapa 27

Combine frações.

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Etapa 27.1

Mova o número negativo para a frente da fração.

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

Etapa 27.2

Combine $\frac{1}{2}$ e ${(a - x)}^{- \frac{1}{2}}$ .

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

Etapa 27.3

Mova ${(a - x)}^{- \frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Etapa 28

De acordo com a regra da soma, a derivada de $a - x$ com relação a $a$ é $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Etapa 29

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d a} [a^{n}]$ é $n a^{n - 1}$ , em que $n = 1$ .

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

Etapa 30

Como $- x$ é constante em relação a $a$ , a derivada de $- x$ em relação a $a$ é $0$ .

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

Etapa 31

Simplifique a expressão.

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Etapa 31.1

Some $1$ e $0$ .

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

Etapa 31.2

Multiplique $\frac{1}{2 {(a - x)}^{\frac{1}{2}}}$ por $1$ .

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

Etapa 32

Simplifique.

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Etapa 32.1

Aplique a propriedade distributiva.

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

Etapa 32.2

Simplifique o numerador.

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Etapa 32.2.1

Expanda $({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} - \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ usando o método FOIL.

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Etapa 32.2.1.1

Aplique a propriedade distributiva.

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

Etapa 32.2.1.2

Aplique a propriedade distributiva.

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

Etapa 32.2.1.3

Aplique a propriedade distributiva.

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

Etapa 32.2.2

Simplifique e combine termos semelhantes.

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Etapa 32.2.2.1

Simplifique cada termo.

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Etapa 32.2.2.1.1

Cancele o fator comum de ${(a + x)}^{\frac{1}{2}}$ .

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Etapa 32.2.2.1.1.1

Fatore ${(a + x)}^{\frac{1}{2}}$ de $2 {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.2.1.1.2

Cancele o fator comum.

Etapa 32.2.2.1.1.3

Reescreva a expressão.

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

Etapa 32.2.2.1.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 32.2.2.1.3

Combine $\frac{1}{2 {(a - x)}^{\frac{1}{2}}}$ e ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.2.1.4

Combine ${(a - x)}^{\frac{1}{2}}$ e $\frac{1}{2 {(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.1.5

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 32.2.2.1.6

Cancele o fator comum de ${(a - x)}^{\frac{1}{2}}$ .

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Etapa 32.2.2.1.6.1

Fatore ${(a - x)}^{\frac{1}{2}}$ de $- {(a - x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.2.1.6.2

Fatore ${(a - x)}^{\frac{1}{2}}$ de $2 {(a - x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.2.1.6.3

Cancele o fator comum.

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

Etapa 32.2.2.1.6.4

Reescreva a expressão.

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

Etapa 32.2.2.2

Para escrever $\frac{1}{2}$ como fração com um denominador comum, multiplique por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.3

Multiplique $\frac{1}{2}$ por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.4

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.2.5

Para escrever $\frac{{(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ como fração com um denominador comum, multiplique por $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.6

Para escrever $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ como fração com um denominador comum, multiplique por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.7

Escreva cada expressão com um denominador comum de $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 32.2.2.7.1

Multiplique $\frac{{(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ por $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.7.2

Multiplique $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.2.7.3

Reordene os fatores de $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.2.8

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.3

Simplifique o numerador.

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Etapa 32.2.3.1

Aplique a propriedade distributiva.

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

Etapa 32.2.3.2

Multiplique ${(a + x)}^{\frac{1}{2}}$ por ${(a + x)}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 32.2.3.2.1

Mova ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.3.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 32.2.3.2.3

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.3.2.4

Some $1$ e $1$ .

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

Etapa 32.2.3.2.5

Divida $2$ por $2$ .

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

Etapa 32.2.3.3

Simplifique $- {(a + x)}^{1}$ .

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

Etapa 32.2.3.4

Aplique a propriedade distributiva.

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

Etapa 32.2.3.5

Multiplique ${(a - x)}^{\frac{1}{2}}$ por ${(a - x)}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 32.2.3.5.1

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 32.2.3.5.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.3.5.3

Some $1$ e $1$ .

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

Etapa 32.2.3.5.4

Divida $2$ por $2$ .

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

Etapa 32.2.3.6

Simplifique ${(a - x)}^{1}$ .

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

Etapa 32.2.3.7

Some $- a$ e $a$ .

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

Etapa 32.2.3.8

Some ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ e $0$ .

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

Etapa 32.2.3.9

Subtraia $x$ de $- x$ .

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

Etapa 32.2.4

Multiplique $- - {(a - x)}^{\frac{1}{2}}$ .

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Etapa 32.2.4.1

Multiplique $- 1$ por $- 1$ .

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

Etapa 32.2.4.2

Multiplique ${(a - x)}^{\frac{1}{2}}$ por $1$ .

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

Etapa 32.2.5

Expanda $(- {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} + \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ usando o método FOIL.

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Etapa 32.2.5.1

Aplique a propriedade distributiva.

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

Etapa 32.2.5.2

Aplique a propriedade distributiva.

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

Etapa 32.2.5.3

Aplique a propriedade distributiva.

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

Etapa 32.2.6

Simplifique e combine termos semelhantes.

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Etapa 32.2.6.1

Simplifique cada termo.

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Etapa 32.2.6.1.1

Cancele o fator comum de ${(a + x)}^{\frac{1}{2}}$ .

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Etapa 32.2.6.1.1.1

Fatore ${(a + x)}^{\frac{1}{2}}$ de $- {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.6.1.1.2

Fatore ${(a + x)}^{\frac{1}{2}}$ de $2 {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.6.1.1.3

Cancele o fator comum.

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

Etapa 32.2.6.1.1.4

Reescreva a expressão.

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

Etapa 32.2.6.1.2

Reescreva $- 1 (\frac{1}{2})$ como $- (\frac{1}{2})$ .

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

Etapa 32.2.6.1.3

Combine $\frac{1}{2 {(a - x)}^{\frac{1}{2}}}$ e ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.6.1.4

Combine ${(a - x)}^{\frac{1}{2}}$ e $\frac{1}{2 {(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.1.5

Cancele o fator comum de ${(a - x)}^{\frac{1}{2}}$ .

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Etapa 32.2.6.1.5.1

Fatore ${(a - x)}^{\frac{1}{2}}$ de $2 {(a - x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.6.1.5.2

Cancele o fator comum.

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

Etapa 32.2.6.1.5.3

Reescreva a expressão.

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

Etapa 32.2.6.2

Para escrever $- \frac{1}{2}$ como fração com um denominador comum, multiplique por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.3

Multiplique $\frac{1}{2}$ por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.4

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.6.5

Para escrever $\frac{- {(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ como fração com um denominador comum, multiplique por $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.6

Para escrever $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ como fração com um denominador comum, multiplique por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.7

Escreva cada expressão com um denominador comum de $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 32.2.6.7.1

Multiplique $\frac{- {(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ por $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.7.2

Multiplique $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ por $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.2.6.7.3

Reordene os fatores de $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.6.8

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.7

Simplifique o numerador.

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Etapa 32.2.7.1

Aplique a propriedade distributiva.

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

Etapa 32.2.7.2

Multiplique ${(a + x)}^{\frac{1}{2}}$ por ${(a + x)}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 32.2.7.2.1

Mova ${(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.7.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 32.2.7.2.3

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.7.2.4

Some $1$ e $1$ .

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

Etapa 32.2.7.2.5

Divida $2$ por $2$ .

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

Etapa 32.2.7.3

Simplifique $- {(a + x)}^{1}$ .

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

Etapa 32.2.7.4

Aplique a propriedade distributiva.

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

Etapa 32.2.7.5

Multiplique ${(a - x)}^{\frac{1}{2}}$ por ${(a - x)}^{\frac{1}{2}}$ somando os expoentes.

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Etapa 32.2.7.5.1

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 32.2.7.5.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.7.5.3

Some $1$ e $1$ .

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

Etapa 32.2.7.5.4

Divida $2$ por $2$ .

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

Etapa 32.2.7.6

Simplifique ${(a - x)}^{1}$ .

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

Etapa 32.2.7.7

Some $- a$ e $a$ .

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

Etapa 32.2.7.8

Some $- {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ e $0$ .

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

Etapa 32.2.7.9

Subtraia $x$ de $- x$ .

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

Etapa 32.2.8

Some $- \frac{1}{2}$ e $\frac{1}{2}$ .

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

Etapa 32.2.9

Some $\frac{{(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x}{2 {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}$ e $0$ .

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

Etapa 32.2.10

Combine os numeradores em relação ao denominador comum.

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

Etapa 32.2.11

Combine os termos opostos em ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x - {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x$ .

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Etapa 32.2.11.1

Subtraia ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ de ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.11.2

Some $- 2 x$ e $0$ .

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

Etapa 32.2.12

Subtraia $2 x$ de $- 2 x$ .

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

Etapa 32.2.13

Fatore $2$ de $- 4 x$ .

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

Etapa 32.2.14

Cancele os fatores comuns.

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Etapa 32.2.14.1

Fatore $2$ de $2 {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}$ .

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

Etapa 32.2.14.2

Cancele o fator comum.

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

Etapa 32.2.14.3

Reescreva a expressão.

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

Etapa 32.2.15

Mova o número negativo para a frente da fração.

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

Etapa 32.3

Combine os termos.

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Etapa 32.3.1

Reescreva $\frac{- \frac{2 x}{{(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2}}$ como um produto.

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

Etapa 32.3.2

Multiplique $\frac{1}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2}}$ por $\frac{2 x}{{(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}$ .

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

Etapa 32.4

Reordene os fatores em $- \frac{2 x}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2} {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}$ .

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