Calculus Examples

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

Step 1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + x}$ as ${(a + x)}^{\frac{1}{2}}$ .

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

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - x}$ as ${(a - x)}^{\frac{1}{2}}$ .

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

Step 1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + x}$ as ${(a + x)}^{\frac{1}{2}}$ .

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

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - x}$ as ${(a - x)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d a} [\frac{f (a)}{g (a)}]$ is $\frac{g (a) \frac{d}{d a} [f (a)] - f (a) \frac{d}{d a} [g (a)]}{{g (a)}^{2}}$ where $f (a) = {(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ and $g (a) = {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ .

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

Step 3

By the Sum Rule, the derivative of ${(a + x)}^{\frac{1}{2}} - {(a - x)}^{\frac{1}{2}}$ with respect to $a$ is $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [- {(a - x)}^{\frac{1}{2}}]$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{\frac{1}{2}}$ and $g (a) = a + x$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{1}$ as $a + x$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 4.3

Replace all occurrences of $u_{1}$ with $a + x$ .

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Differentiate.

Tap for more steps...

Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

Combine fractions.

Tap for more steps...

Step 9.2.1

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

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

Step 9.2.2

Move ${(a + x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.3

By the Sum Rule, the derivative of $a + x$ with respect to $a$ is $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Step 9.4

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 1$ .

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

Step 9.5

Since $x$ is constant with respect to $a$ , the derivative of $x$ with respect to $a$ is $0$ .

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

Step 9.6

Simplify the expression.

Tap for more steps...

Step 9.6.1

Add $1$ and $0$ .

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

Step 9.6.2

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

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

Step 9.7

Since $- 1$ is constant with respect to $a$ , the derivative of $- {(a - x)}^{\frac{1}{2}}$ with respect to $a$ is $- \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Step 10

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{\frac{1}{2}}$ and $g (a) = a - x$ .

Tap for more steps...

Step 10.1

To apply the Chain Rule, set $u_{2}$ as $a - x$ .

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

Step 10.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 10.3

Replace all occurrences of $u_{2}$ with $a - x$ .

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

Step 11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12

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

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Simplify the numerator.

Tap for more steps...

Step 14.1

Multiply $- 1$ by $2$ .

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

Step 14.2

Subtract $2$ from $1$ .

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

Step 15

Differentiate.

Tap for more steps...

Step 15.1

Move the negative in front of the fraction.

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

Step 15.2

Combine fractions.

Tap for more steps...

Step 15.2.1

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

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

Step 15.2.2

Move ${(a - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 15.3

By the Sum Rule, the derivative of $a - x$ with respect to $a$ is $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Step 15.4

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 1$ .

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

Step 15.5

Since $- x$ is constant with respect to $a$ , the derivative of $- x$ with respect to $a$ is $0$ .

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

Step 15.6

Simplify the expression.

Tap for more steps...

Step 15.6.1

Add $1$ and $0$ .

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

Step 15.6.2

Multiply $- 1$ by $1$ .

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

Step 15.7

By the Sum Rule, the derivative of ${(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}$ with respect to $a$ is $\frac{d}{d a} [{(a + x)}^{\frac{1}{2}}] + \frac{d}{d a} [{(a - x)}^{\frac{1}{2}}]$ .

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

Step 16

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{\frac{1}{2}}$ and $g (a) = a + x$ .

Tap for more steps...

Step 16.1

To apply the Chain Rule, set $u_{3}$ as $a + x$ .

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

Step 16.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 16.3

Replace all occurrences of $u_{3}$ with $a + x$ .

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

Step 17

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 18

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

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Simplify the numerator.

Tap for more steps...

Step 20.1

Multiply $- 1$ by $2$ .

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

Step 20.2

Subtract $2$ from $1$ .

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

Step 21

Differentiate.

Tap for more steps...

Step 21.1

Move the negative in front of the fraction.

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

Step 21.2

Combine fractions.

Tap for more steps...

Step 21.2.1

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

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

Step 21.2.2

Move ${(a + x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 21.3

By the Sum Rule, the derivative of $a + x$ with respect to $a$ is $\frac{d}{d a} [a] + \frac{d}{d a} [x]$ .

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

Step 21.4

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 1$ .

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

Step 21.5

Since $x$ is constant with respect to $a$ , the derivative of $x$ with respect to $a$ is $0$ .

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

Step 21.6

Simplify the expression.

Tap for more steps...

Step 21.6.1

Add $1$ and $0$ .

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

Step 21.6.2

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

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

Step 22

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{\frac{1}{2}}$ and $g (a) = a - x$ .

Tap for more steps...

Step 22.1

To apply the Chain Rule, set $u_{4}$ as $a - x$ .

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

Step 22.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 22.3

Replace all occurrences of $u_{4}$ with $a - x$ .

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

Step 23

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 24

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

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

Step 25

Combine the numerators over the common denominator.

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

Step 26

Simplify the numerator.

Tap for more steps...

Step 26.1

Multiply $- 1$ by $2$ .

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

Step 26.2

Subtract $2$ from $1$ .

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

Step 27

Combine fractions.

Tap for more steps...

Step 27.1

Move the negative in front of the fraction.

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

Step 27.2

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

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

Step 27.3

Move ${(a - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 28

By the Sum Rule, the derivative of $a - x$ with respect to $a$ is $\frac{d}{d a} [a] + \frac{d}{d a} [- x]$ .

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

Step 29

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 1$ .

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

Step 30

Since $- x$ is constant with respect to $a$ , the derivative of $- x$ with respect to $a$ is $0$ .

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

Step 31

Simplify the expression.

Tap for more steps...

Step 31.1

Add $1$ and $0$ .

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

Step 31.2

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

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

Step 32

Simplify.

Tap for more steps...

Step 32.1

Apply the distributive property.

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

Step 32.2

Simplify the numerator.

Tap for more steps...

Step 32.2.1

Expand $({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} - \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ using the FOIL Method.

Tap for more steps...

Step 32.2.1.1

Apply the distributive property.

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

Step 32.2.1.2

Apply the distributive property.

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

Step 32.2.1.3

Apply the distributive property.

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

Step 32.2.2

Simplify and combine like terms.

Tap for more steps...

Step 32.2.2.1

Simplify each term.

Tap for more steps...

Step 32.2.2.1.1

Cancel the common factor of ${(a + x)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 32.2.2.1.1.1

Factor ${(a + x)}^{\frac{1}{2}}$ out of $2 {(a + x)}^{\frac{1}{2}}$ .

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

Step 32.2.2.1.1.2

Cancel the common factor.

Step 32.2.2.1.1.3

Rewrite the expression.

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

Step 32.2.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 32.2.2.1.3

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

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

Step 32.2.2.1.4

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

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

Step 32.2.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 32.2.2.1.6

Cancel the common factor of ${(a - x)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 32.2.2.1.6.1

Factor ${(a - x)}^{\frac{1}{2}}$ out of $- {(a - x)}^{\frac{1}{2}}$ .

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

Step 32.2.2.1.6.2

Factor ${(a - x)}^{\frac{1}{2}}$ out of $2 {(a - x)}^{\frac{1}{2}}$ .

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

Step 32.2.2.1.6.3

Cancel the common factor.

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

Step 32.2.2.1.6.4

Rewrite the expression.

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

Step 32.2.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Step 32.2.2.3

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

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

Step 32.2.2.4

Combine the numerators over the common denominator.

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

Step 32.2.2.5

To write $\frac{{(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Step 32.2.2.6

To write $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Step 32.2.2.7

Write each expression with a common denominator of $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 32.2.2.7.1

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

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

Step 32.2.2.7.2

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

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

Step 32.2.2.7.3

Reorder the factors of $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Step 32.2.2.8

Combine the numerators over the common denominator.

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

Step 32.2.3

Simplify the numerator.

Tap for more steps...

Step 32.2.3.1

Apply the distributive property.

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

Step 32.2.3.2

Multiply ${(a + x)}^{\frac{1}{2}}$ by ${(a + x)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 32.2.3.2.1

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

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

Step 32.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 32.2.3.2.3

Combine the numerators over the common denominator.

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

Step 32.2.3.2.4

Add $1$ and $1$ .

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

Step 32.2.3.2.5

Divide $2$ by $2$ .

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

Step 32.2.3.3

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

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

Step 32.2.3.4

Apply the distributive property.

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

Step 32.2.3.5

Multiply ${(a - x)}^{\frac{1}{2}}$ by ${(a - x)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 32.2.3.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 32.2.3.5.2

Combine the numerators over the common denominator.

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

Step 32.2.3.5.3

Add $1$ and $1$ .

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

Step 32.2.3.5.4

Divide $2$ by $2$ .

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

Step 32.2.3.6

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

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

Step 32.2.3.7

Add $- a$ and $a$ .

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

Step 32.2.3.8

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

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

Step 32.2.3.9

Subtract $x$ from $- x$ .

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

Step 32.2.4

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

Tap for more steps...

Step 32.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 32.2.4.2

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

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

Step 32.2.5

Expand $(- {(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}}) (\frac{1}{2 {(a + x)}^{\frac{1}{2}}} + \frac{1}{2 {(a - x)}^{\frac{1}{2}}})$ using the FOIL Method.

Tap for more steps...

Step 32.2.5.1

Apply the distributive property.

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

Step 32.2.5.2

Apply the distributive property.

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

Step 32.2.5.3

Apply the distributive property.

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

Step 32.2.6

Simplify and combine like terms.

Tap for more steps...

Step 32.2.6.1

Simplify each term.

Tap for more steps...

Step 32.2.6.1.1

Cancel the common factor of ${(a + x)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 32.2.6.1.1.1

Factor ${(a + x)}^{\frac{1}{2}}$ out of $- {(a + x)}^{\frac{1}{2}}$ .

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

Step 32.2.6.1.1.2

Factor ${(a + x)}^{\frac{1}{2}}$ out of $2 {(a + x)}^{\frac{1}{2}}$ .

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

Step 32.2.6.1.1.3

Cancel the common factor.

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

Step 32.2.6.1.1.4

Rewrite the expression.

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

Step 32.2.6.1.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

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

Step 32.2.6.1.3

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

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

Step 32.2.6.1.4

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

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

Step 32.2.6.1.5

Cancel the common factor of ${(a - x)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 32.2.6.1.5.1

Factor ${(a - x)}^{\frac{1}{2}}$ out of $2 {(a - x)}^{\frac{1}{2}}$ .

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

Step 32.2.6.1.5.2

Cancel the common factor.

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

Step 32.2.6.1.5.3

Rewrite the expression.

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

Step 32.2.6.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Step 32.2.6.3

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

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

Step 32.2.6.4

Combine the numerators over the common denominator.

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

Step 32.2.6.5

To write $\frac{- {(a - x)}^{\frac{1}{2}} - {(a + x)}^{\frac{1}{2}}}{2 {(a - x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(a + x)}^{\frac{1}{2}}}{{(a + x)}^{\frac{1}{2}}}$ .

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

Step 32.2.6.6

To write $\frac{{(a - x)}^{\frac{1}{2}}}{2 {(a + x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(a - x)}^{\frac{1}{2}}}{{(a - x)}^{\frac{1}{2}}}$ .

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

Step 32.2.6.7

Write each expression with a common denominator of $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 32.2.6.7.1

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

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

Step 32.2.6.7.2

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

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

Step 32.2.6.7.3

Reorder the factors of $2 {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}}$ .

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

Step 32.2.6.8

Combine the numerators over the common denominator.

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

Step 32.2.7

Simplify the numerator.

Tap for more steps...

Step 32.2.7.1

Apply the distributive property.

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

Step 32.2.7.2

Multiply ${(a + x)}^{\frac{1}{2}}$ by ${(a + x)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 32.2.7.2.1

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

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

Step 32.2.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 32.2.7.2.3

Combine the numerators over the common denominator.

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

Step 32.2.7.2.4

Add $1$ and $1$ .

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

Step 32.2.7.2.5

Divide $2$ by $2$ .

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

Step 32.2.7.3

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

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

Step 32.2.7.4

Apply the distributive property.

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

Step 32.2.7.5

Multiply ${(a - x)}^{\frac{1}{2}}$ by ${(a - x)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 32.2.7.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 32.2.7.5.2

Combine the numerators over the common denominator.

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

Step 32.2.7.5.3

Add $1$ and $1$ .

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

Step 32.2.7.5.4

Divide $2$ by $2$ .

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

Step 32.2.7.6

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

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

Step 32.2.7.7

Add $- a$ and $a$ .

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

Step 32.2.7.8

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

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

Step 32.2.7.9

Subtract $x$ from $- x$ .

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

Step 32.2.8

Add $- \frac{1}{2}$ and $\frac{1}{2}$ .

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

Step 32.2.9

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

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

Step 32.2.10

Combine the numerators over the common denominator.

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

Step 32.2.11

Combine the opposite terms in ${(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x - {(a - x)}^{\frac{1}{2}} {(a + x)}^{\frac{1}{2}} - 2 x$ .

Tap for more steps...

Step 32.2.11.1

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

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

Step 32.2.11.2

Add $- 2 x$ and $0$ .

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

Step 32.2.12

Subtract $2 x$ from $- 2 x$ .

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

Step 32.2.13

Factor $2$ out of $- 4 x$ .

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

Step 32.2.14

Cancel the common factors.

Tap for more steps...

Step 32.2.14.1

Factor $2$ out of $2 {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}$ .

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

Step 32.2.14.2

Cancel the common factor.

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

Step 32.2.14.3

Rewrite the expression.

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

Step 32.2.15

Move the negative in front of the fraction.

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

Step 32.3

Combine terms.

Tap for more steps...

Step 32.3.1

Rewrite $\frac{- \frac{2 x}{{(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2}}$ as a product.

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

Step 32.3.2

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

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

Step 32.4

Reorder factors in $- \frac{2 x}{{({(a + x)}^{\frac{1}{2}} + {(a - x)}^{\frac{1}{2}})}^{2} {(a + x)}^{\frac{1}{2}} {(a - x)}^{\frac{1}{2}}}$ .

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