Calcolo Esempi

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

Passaggio 1

Applica le regole di base degli esponenti.

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Passaggio 1.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ per riscrivere $\sqrt{a + x}$ come ${(a + x)}^{\frac{1}{2}}$ .

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

Passaggio 1.2

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ per riscrivere $\sqrt{a - x}$ come ${(a - x)}^{\frac{1}{2}}$ .

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

Passaggio 1.3

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ per riscrivere $\sqrt{a + x}$ come ${(a + x)}^{\frac{1}{2}}$ .

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

Passaggio 1.4

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ per riscrivere $\sqrt{a - x}$ come ${(a - x)}^{\frac{1}{2}}$ .

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

Passaggio 2

Differenzia usando la regola del quoziente, che indica che $\frac{d}{d a} [\frac{f (a)}{g (a)}]$ è $\frac{g (a) \frac{d}{d a} [f (a)] - f (a) \frac{d}{d a} [g (a)]}{{g (a)}^{2}}$ dove $f (a) = {(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ e $g (a) = {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ .

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

Passaggio 3

Secondo la regola della somma, la derivata di ${(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ rispetto a $a$ è $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [- {(a - x)}^{\frac{1}{2}}]$ .

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

Passaggio 4

Differenzia usando la regola della catena, che indica che $\frac{d}{d a} [f (g (a))]$ è $f' (g (a)) g' (a)$ dove $f (a) = a^{\frac{1}{2}}$ e $g (a) = a + x$ .

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Passaggio 4.1

Per applicare la regola della catena, imposta $u_{1}$ come $a + x$ .

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

Passaggio 4.2

Differenzia usando la regola di potenza, che indica che $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ è $n {u_{1}}^{n - 1}$ dove $n = \frac{1}{2}$ .

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

Passaggio 4.3

Sostituisci tutte le occorrenze di $u_{1}$ con $a + x$ .

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

Passaggio 5

Per scrivere $- 1$ come una frazione con un comune denominatore, moltiplicala per $\frac{2}{2}$ .

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

Passaggio 6

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

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

Passaggio 7

Riduci i numeratori su un comune denominatore.

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

Passaggio 8

Semplifica il numeratore.

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Passaggio 8.1

Moltiplica $- 1$ per $2$ .

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

Passaggio 8.2

Sottrai $2$ da $1$ .

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

Passaggio 9

Differenzia.

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Passaggio 9.1

Sposta il negativo davanti alla frazione.

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

Passaggio 9.2

Riduci le frazioni.

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Passaggio 9.2.1

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

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

Passaggio 9.2.2

Sposta ${(a + x)}^{- \frac{1}{2}}$ al denominatore usando la regola dell'esponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Passaggio 9.3

Secondo la regola della somma, la derivata di $a + x$ rispetto a $a$ è $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Passaggio 9.4

Differenzia usando la regola di potenza, che indica che $\frac{d}{d a} [a^{n}]$ è $n a^{n - 1}$ dove $n = 1$ .

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

Passaggio 9.5

Poiché $x$ è costante rispetto a $a$ , la derivata di $x$ rispetto a $a$ è $0$ .

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

Passaggio 9.6

Semplifica l'espressione.

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Passaggio 9.6.1

Somma $1$ e $0$ .

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

Passaggio 9.6.2

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

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

Passaggio 9.7

Poiché $- 1$ è costante rispetto a $a$ , la derivata di $- {(a - x)}^{\frac{1}{2}}$ rispetto a $a$ è $- \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Passaggio 10

Differenzia usando la regola della catena, che indica che $\frac{d}{d a} [f (g (a))]$ è $f' (g (a)) g' (a)$ dove $f (a) = a^{\frac{1}{2}}$ e $g (a) = a - x$ .

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Passaggio 10.1

Per applicare la regola della catena, imposta $u_{2}$ come $a - x$ .

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

Passaggio 10.2

Differenzia usando la regola di potenza, che indica che $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ è $n {u_{2}}^{n - 1}$ dove $n = \frac{1}{2}$ .

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

Passaggio 10.3

Sostituisci tutte le occorrenze di $u_{2}$ con $a - x$ .

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

Passaggio 11

Per scrivere $- 1$ come una frazione con un comune denominatore, moltiplicala per $\frac{2}{2}$ .

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

Passaggio 12

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

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

Passaggio 13

Riduci i numeratori su un comune denominatore.

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

Passaggio 14

Semplifica il numeratore.

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Passaggio 14.1

Moltiplica $- 1$ per $2$ .

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

Passaggio 14.2

Sottrai $2$ da $1$ .

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

Passaggio 15

Differenzia.

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Passaggio 15.1

Sposta il negativo davanti alla frazione.

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

Passaggio 15.2

Riduci le frazioni.

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Passaggio 15.2.1

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

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

Passaggio 15.2.2

Sposta ${(a - x)}^{- \frac{1}{2}}$ al denominatore usando la regola dell'esponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Passaggio 15.3

Secondo la regola della somma, la derivata di $a - x$ rispetto a $a$ è $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Passaggio 15.4

Differenzia usando la regola di potenza, che indica che $\frac{d}{d a} [a^{n}]$ è $n a^{n - 1}$ dove $n = 1$ .

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

Passaggio 15.5

Poiché $- x$ è costante rispetto a $a$ , la derivata di $- x$ rispetto a $a$ è $0$ .

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

Passaggio 15.6

Semplifica l'espressione.

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Passaggio 15.6.1

Somma $1$ e $0$ .

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

Passaggio 15.6.2

Moltiplica $- 1$ per $1$ .

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

Passaggio 15.7

Secondo la regola della somma, la derivata di ${(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ rispetto a $a$ è $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Passaggio 16

Differenzia usando la regola della catena, che indica che $\frac{d}{d a} [f (g (a))]$ è $f' (g (a)) g' (a)$ dove $f (a) = a^{\frac{1}{2}}$ e $g (a) = a + x$ .

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Passaggio 16.1

Per applicare la regola della catena, imposta $u_{3}$ come $a + x$ .

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

Passaggio 16.2

Differenzia usando la regola di potenza, che indica che $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ è $n {u_{3}}^{n - 1}$ dove $n = \frac{1}{2}$ .

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

Passaggio 16.3

Sostituisci tutte le occorrenze di $u_{3}$ con $a + x$ .

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

Passaggio 17

Per scrivere $- 1$ come una frazione con un comune denominatore, moltiplicala per $\frac{2}{2}$ .

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

Passaggio 18

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

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

Passaggio 19

Riduci i numeratori su un comune denominatore.

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

Passaggio 20

Semplifica il numeratore.

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Passaggio 20.1

Moltiplica $- 1$ per $2$ .

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

Passaggio 20.2

Sottrai $2$ da $1$ .

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

Passaggio 21

Differenzia.

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Passaggio 21.1

Sposta il negativo davanti alla frazione.

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

Passaggio 21.2

Riduci le frazioni.

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Passaggio 21.2.1

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

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

Passaggio 21.2.2

Sposta ${(a + x)}^{- \frac{1}{2}}$ al denominatore usando la regola dell'esponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Passaggio 21.3

Secondo la regola della somma, la derivata di $a + x$ rispetto a $a$ è $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Passaggio 21.4

Differenzia usando la regola di potenza, che indica che $\frac{d}{d a} [a^{n}]$ è $n a^{n - 1}$ dove $n = 1$ .

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

Passaggio 21.5

Poiché $x$ è costante rispetto a $a$ , la derivata di $x$ rispetto a $a$ è $0$ .

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

Passaggio 21.6

Semplifica l'espressione.

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Passaggio 21.6.1

Somma $1$ e $0$ .

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

Passaggio 21.6.2

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

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

Passaggio 22

Differenzia usando la regola della catena, che indica che $\frac{d}{d a} [f (g (a))]$ è $f' (g (a)) g' (a)$ dove $f (a) = a^{\frac{1}{2}}$ e $g (a) = a - x$ .

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Passaggio 22.1

Per applicare la regola della catena, imposta $u_{4}$ come $a - x$ .

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

Passaggio 22.2

Differenzia usando la regola di potenza, che indica che $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ è $n {u_{4}}^{n - 1}$ dove $n = \frac{1}{2}$ .

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

Passaggio 22.3

Sostituisci tutte le occorrenze di $u_{4}$ con $a - x$ .

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

Passaggio 23

Per scrivere $- 1$ come una frazione con un comune denominatore, moltiplicala per $\frac{2}{2}$ .

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

Passaggio 24

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

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

Passaggio 25

Riduci i numeratori su un comune denominatore.

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

Passaggio 26

Semplifica il numeratore.

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Passaggio 26.1

Moltiplica $- 1$ per $2$ .

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

Passaggio 26.2

Sottrai $2$ da $1$ .

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

Passaggio 27

Riduci le frazioni.

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Passaggio 27.1

Sposta il negativo davanti alla frazione.

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

Passaggio 27.2

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

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

Passaggio 27.3

Sposta ${(a - x)}^{- \frac{1}{2}}$ al denominatore usando la regola dell'esponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

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

Passaggio 28

Secondo la regola della somma, la derivata di $a - x$ rispetto a $a$ è $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Passaggio 29

Differenzia usando la regola di potenza, che indica che $\frac{d}{d a} [a^{n}]$ è $n a^{n - 1}$ dove $n = 1$ .

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

Passaggio 30

Poiché $- x$ è costante rispetto a $a$ , la derivata di $- x$ rispetto a $a$ è $0$ .

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

Passaggio 31

Semplifica l'espressione.

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Passaggio 31.1

Somma $1$ e $0$ .

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

Passaggio 31.2

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

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

Passaggio 32

Semplifica.

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Passaggio 32.1

Applica la proprietà distributiva.

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

Passaggio 32.2

Semplifica il numeratore.

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Passaggio 32.2.1

Espandi $({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} - \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ usando il metodo FOIL.

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Passaggio 32.2.1.1

Applica la proprietà distributiva.

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

Passaggio 32.2.1.2

Applica la proprietà distributiva.

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

Passaggio 32.2.1.3

Applica la proprietà distributiva.

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

Passaggio 32.2.2

Semplifica e combina i termini simili.

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Passaggio 32.2.2.1

Semplifica ciascun termine.

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Passaggio 32.2.2.1.1

Elimina il fattore comune di ${(a + x)}^{\frac{1}{2}}$ .

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Passaggio 32.2.2.1.1.1

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

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

Passaggio 32.2.2.1.1.2

Elimina il fattore comune.

Passaggio 32.2.2.1.1.3

Riscrivi l'espressione.

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

Passaggio 32.2.2.1.2

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

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

Passaggio 32.2.2.1.3

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

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

Passaggio 32.2.2.1.4

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

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

Passaggio 32.2.2.1.5

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

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

Passaggio 32.2.2.1.6

Elimina il fattore comune di ${(a - x)}^{\frac{1}{2}}$ .

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Passaggio 32.2.2.1.6.1

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

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

Passaggio 32.2.2.1.6.2

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

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

Passaggio 32.2.2.1.6.3

Elimina il fattore comune.

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

Passaggio 32.2.2.1.6.4

Riscrivi l'espressione.

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

Passaggio 32.2.2.2

Per scrivere $\frac{1}{2}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.2.3

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

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

Passaggio 32.2.2.4

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.2.5

Per scrivere $\frac{{(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.2.6

Per scrivere $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.2.7

Scrivi ogni espressione con un comune denominatore di $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , moltiplicando ciascuna per il fattore appropriato di $1$ .

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Passaggio 32.2.2.7.1

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

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

Passaggio 32.2.2.7.2

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

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

Passaggio 32.2.2.7.3

Riordina i fattori di $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Passaggio 32.2.2.8

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.3

Semplifica il numeratore.

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Passaggio 32.2.3.1

Applica la proprietà distributiva.

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

Passaggio 32.2.3.2

Moltiplica ${(a + x)}^{\frac{1}{2}}$ per ${(a + x)}^{\frac{1}{2}}$ sommando gli esponenti.

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Passaggio 32.2.3.2.1

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

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

Passaggio 32.2.3.2.2

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 32.2.3.2.3

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.3.2.4

Somma $1$ e $1$ .

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

Passaggio 32.2.3.2.5

Dividi $2$ per $2$ .

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

Passaggio 32.2.3.3

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

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

Passaggio 32.2.3.4

Applica la proprietà distributiva.

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

Passaggio 32.2.3.5

Moltiplica ${(a - x)}^{\frac{1}{2}}$ per ${(a - x)}^{\frac{1}{2}}$ sommando gli esponenti.

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Passaggio 32.2.3.5.1

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 32.2.3.5.2

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.3.5.3

Somma $1$ e $1$ .

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

Passaggio 32.2.3.5.4

Dividi $2$ per $2$ .

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

Passaggio 32.2.3.6

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

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

Passaggio 32.2.3.7

Somma $- a$ e $a$ .

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

Passaggio 32.2.3.8

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

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

Passaggio 32.2.3.9

Sottrai $x$ da $- x$ .

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

Passaggio 32.2.4

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

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Passaggio 32.2.4.1

Moltiplica $- 1$ per $- 1$ .

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

Passaggio 32.2.4.2

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

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

Passaggio 32.2.5

Espandi $(- {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} + \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ usando il metodo FOIL.

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Passaggio 32.2.5.1

Applica la proprietà distributiva.

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

Passaggio 32.2.5.2

Applica la proprietà distributiva.

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

Passaggio 32.2.5.3

Applica la proprietà distributiva.

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

Passaggio 32.2.6

Semplifica e combina i termini simili.

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Passaggio 32.2.6.1

Semplifica ciascun termine.

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Passaggio 32.2.6.1.1

Elimina il fattore comune di ${(a + x)}^{\frac{1}{2}}$ .

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Passaggio 32.2.6.1.1.1

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

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

Passaggio 32.2.6.1.1.2

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

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

Passaggio 32.2.6.1.1.3

Elimina il fattore comune.

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

Passaggio 32.2.6.1.1.4

Riscrivi l'espressione.

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

Passaggio 32.2.6.1.2

Riscrivi $- 1 (\frac{1}{2})$ come $- (\frac{1}{2})$ .

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

Passaggio 32.2.6.1.3

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

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

Passaggio 32.2.6.1.4

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

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

Passaggio 32.2.6.1.5

Elimina il fattore comune di ${(a - x)}^{\frac{1}{2}}$ .

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Passaggio 32.2.6.1.5.1

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

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

Passaggio 32.2.6.1.5.2

Elimina il fattore comune.

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

Passaggio 32.2.6.1.5.3

Riscrivi l'espressione.

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

Passaggio 32.2.6.2

Per scrivere $- \frac{1}{2}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.6.3

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

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

Passaggio 32.2.6.4

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.6.5

Per scrivere $\frac{- {(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.6.6

Per scrivere $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ come una frazione con un comune denominatore, moltiplicala per $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Passaggio 32.2.6.7

Scrivi ogni espressione con un comune denominatore di $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , moltiplicando ciascuna per il fattore appropriato di $1$ .

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Passaggio 32.2.6.7.1

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

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

Passaggio 32.2.6.7.2

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

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

Passaggio 32.2.6.7.3

Riordina i fattori di $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Passaggio 32.2.6.8

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.7

Semplifica il numeratore.

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Passaggio 32.2.7.1

Applica la proprietà distributiva.

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

Passaggio 32.2.7.2

Moltiplica ${(a + x)}^{\frac{1}{2}}$ per ${(a + x)}^{\frac{1}{2}}$ sommando gli esponenti.

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Passaggio 32.2.7.2.1

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

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

Passaggio 32.2.7.2.2

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 32.2.7.2.3

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.7.2.4

Somma $1$ e $1$ .

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

Passaggio 32.2.7.2.5

Dividi $2$ per $2$ .

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

Passaggio 32.2.7.3

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

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

Passaggio 32.2.7.4

Applica la proprietà distributiva.

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

Passaggio 32.2.7.5

Moltiplica ${(a - x)}^{\frac{1}{2}}$ per ${(a - x)}^{\frac{1}{2}}$ sommando gli esponenti.

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Passaggio 32.2.7.5.1

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 32.2.7.5.2

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.7.5.3

Somma $1$ e $1$ .

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

Passaggio 32.2.7.5.4

Dividi $2$ per $2$ .

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

Passaggio 32.2.7.6

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

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

Passaggio 32.2.7.7

Somma $- a$ e $a$ .

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

Passaggio 32.2.7.8

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

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

Passaggio 32.2.7.9

Sottrai $x$ da $- x$ .

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

Passaggio 32.2.8

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

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

Passaggio 32.2.9

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

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

Passaggio 32.2.10

Riduci i numeratori su un comune denominatore.

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

Passaggio 32.2.11

Combina i termini opposti in ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x - {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x$ .

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Passaggio 32.2.11.1

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

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

Passaggio 32.2.11.2

Somma $- 2 x$ e $0$ .

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

Passaggio 32.2.12

Sottrai $2 x$ da $- 2 x$ .

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

Passaggio 32.2.13

Scomponi $2$ da $- 4 x$ .

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

Passaggio 32.2.14

Elimina i fattori comuni.

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Passaggio 32.2.14.1

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

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

Passaggio 32.2.14.2

Elimina il fattore comune.

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

Passaggio 32.2.14.3

Riscrivi l'espressione.

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

Passaggio 32.2.15

Sposta il negativo davanti alla frazione.

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

Passaggio 32.3

Raccogli i termini.

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Passaggio 32.3.1

Riscrivi $\frac{- \frac{2 x}{{(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2}}$ come un prodotto.

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

Passaggio 32.3.2

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

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

Passaggio 32.4

Riordina i fattori in $- \frac{2 x}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2} {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}$ .

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