Calculus Examples

$y \sqrt{x + 4} = x y + 8$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$y {(x + 4)}^{\frac{1}{2}} = x y + 8$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y {(x + 4)}^{\frac{1}{2}}) = \frac{d}{d x} (x y + 8)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = y$ and $g (x) = {(x + 4)}^{\frac{1}{2}}$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 4$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x + 4$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x + 4$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $1$ .

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

Step 3.7

Combine fractions.

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Step 3.7.1

Move the negative in front of the fraction.

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

Step 3.7.2

Combine $\frac{1}{2}$ and ${(x + 4)}^{- \frac{1}{2}}$ .

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

Step 3.7.3

Move ${(x + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.7.4

Combine $\frac{1}{2 {(x + 4)}^{\frac{1}{2}}}$ and $y$ .

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

Step 3.8

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

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

Step 3.9

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

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

Step 3.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{y}{2 {(x + 4)}^{\frac{1}{2}}} (1 + 0) + {(x + 4)}^{\frac{1}{2}} \frac{d}{d x} [y]$

Step 3.11

Simplify the expression.

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Step 3.11.1

Add $1$ and $0$ .

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

Step 3.11.2

Multiply $\frac{y}{2 {(x + 4)}^{\frac{1}{2}}}$ by $1$ .

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

Step 3.12

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{y}{2 {(x + 4)}^{\frac{1}{2}}} + {(x + 4)}^{\frac{1}{2}} y'$

Step 3.13

Reorder terms.

${(x + 4)}^{\frac{1}{2}} y' + \frac{y}{2 {(x + 4)}^{\frac{1}{2}}}$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $x y + 8$ with respect to $x$ is $\frac{d}{d x} [x y] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [x y] + \frac{d}{d x} [8]$

Step 4.2

Evaluate $\frac{d}{d x} [x y]$ .

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Step 4.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$x \frac{d}{d x} [y] + y \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 4.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x y' + y \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 4.2.3

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

$x y' + y \cdot 1 + \frac{d}{d x} [8]$

Step 4.2.4

Multiply $y$ by $1$ .

$x y' + y + \frac{d}{d x} [8]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$x y' + y + 0$

Step 4.3.2

Add $x y' + y$ and $0$ .

$x y' + y$

Step 5

Reform the equation by setting the left side equal to the right side.

${(x + 4)}^{\frac{1}{2}} y' + \frac{y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6

Solve for $y'$ .

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Step 6.1

Simplify ${(x + 4)}^{\frac{1}{2}} y' + \frac{y}{2 {(x + 4)}^{\frac{1}{2}}}$ .

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Step 6.1.1

To write ${(x + 4)}^{\frac{1}{2}} y'$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}}$ .

${(x + 4)}^{\frac{1}{2}} y' \cdot \frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} + \frac{y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Combine ${(x + 4)}^{\frac{1}{2}} y'$ and $\frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}}$ .

$\frac{{(x + 4)}^{\frac{1}{2}} y' (2 {(x + 4)}^{\frac{1}{2}})}{2 {(x + 4)}^{\frac{1}{2}}} + \frac{y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.2.2

Combine the numerators over the common denominator.

$\frac{{(x + 4)}^{\frac{1}{2}} y' (2 {(x + 4)}^{\frac{1}{2}}) + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{2 {(x + 4)}^{\frac{1}{2}} y' {(x + 4)}^{\frac{1}{2}} + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.2

Multiply ${(x + 4)}^{\frac{1}{2}}$ by ${(x + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 6.1.3.2.1

Move ${(x + 4)}^{\frac{1}{2}}$ .

$\frac{2 ({(x + 4)}^{\frac{1}{2}} {(x + 4)}^{\frac{1}{2}}) y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.2.2

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

$\frac{2 {(x + 4)}^{\frac{1}{2} + \frac{1}{2}} y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.2.3

Combine the numerators over the common denominator.

$\frac{2 {(x + 4)}^{\frac{1 + 1}{2}} y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.2.4

Add $1$ and $1$ .

$\frac{2 {(x + 4)}^{\frac{2}{2}} y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.2.5

Divide $2$ by $2$ .

$\frac{2 {(x + 4)}^{1} y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.3

Simplify $2 {(x + 4)}^{1} y'$ .

$\frac{2 (x + 4) y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.4

Apply the distributive property.

$\frac{(2 x + 2 \cdot 4) y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.5

Multiply $2$ by $4$ .

$\frac{(2 x + 8) y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.1.3.6

Apply the distributive property.

$\frac{2 x y' + 8 y' + y}{2 {(x + 4)}^{\frac{1}{2}}} = x y' + y$

Step 6.2

Move all terms containing $y'$ to the left side of the equation.

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Step 6.2.1

Subtract $x y'$ from both sides of the equation.

$\frac{2 x y' + 8 y' + y}{2 {(x + 4)}^{\frac{1}{2}}} - x y' = y$

Step 6.2.2

To write $- x y'$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}}$ .

$\frac{2 x y' + 8 y' + y}{2 {(x + 4)}^{\frac{1}{2}}} - x y' \cdot \frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} = y$

Step 6.2.3

Combine $- x y'$ and $\frac{2 {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}}$ .

$\frac{2 x y' + 8 y' + y}{2 {(x + 4)}^{\frac{1}{2}}} + \frac{- x y' (2 {(x + 4)}^{\frac{1}{2}})}{2 {(x + 4)}^{\frac{1}{2}}} = y$

Step 6.2.4

Combine the numerators over the common denominator.

$\frac{2 x y' + 8 y' + y - x y' (2 {(x + 4)}^{\frac{1}{2}})}{2 {(x + 4)}^{\frac{1}{2}}} = y$

Step 6.2.5

Multiply $2$ by $- 1$ .

$\frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} = y$

Step 6.3

Multiply both sides by $2 {(x + 4)}^{\frac{1}{2}}$ .

$\frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} (2 {(x + 4)}^{\frac{1}{2}}) = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4

Simplify.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Simplify $\frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} (2 {(x + 4)}^{\frac{1}{2}})$ .

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Step 6.4.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.1.1.2.1

Cancel the common factor.

$2 \frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{2 {(x + 4)}^{\frac{1}{2}}} {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.2.2

Rewrite the expression.

$\frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{{(x + 4)}^{\frac{1}{2}}} {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.3

Cancel the common factor of ${(x + 4)}^{\frac{1}{2}}$ .

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Step 6.4.1.1.3.1

Cancel the common factor.

$\frac{2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}}}{{(x + 4)}^{\frac{1}{2}}} {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.3.2

Rewrite the expression.

$2 x y' + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.4

Reorder.

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Step 6.4.1.1.4.1

Move $x$ .

$2 y' x + 8 y' + y - 2 x y' {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.4.2

Move $x$ .

$2 y' x + 8 y' + y - 2 y' x {(x + 4)}^{\frac{1}{2}} = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.1.1.4.3

Move $y$ .

$2 y' x + 8 y' - 2 y' x {(x + 4)}^{\frac{1}{2}} + y = y (2 {(x + 4)}^{\frac{1}{2}})$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Rewrite using the commutative property of multiplication.

$2 y' x + 8 y' - 2 y' x {(x + 4)}^{\frac{1}{2}} + y = 2 y {(x + 4)}^{\frac{1}{2}}$

Step 6.5

Solve for $y'$ .

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Step 6.5.1

Subtract $y$ from both sides of the equation.

$2 y' x + 8 y' - 2 y' x {(x + 4)}^{\frac{1}{2}} = 2 y {(x + 4)}^{\frac{1}{2}} - y$

Step 6.5.2

Factor $2 y'$ out of $2 y' x + 8 y' - 2 y' x {(x + 4)}^{\frac{1}{2}}$ .

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Step 6.5.2.1

Factor $2 y'$ out of $2 y' x$ .

$2 y' x + 8 y' - 2 y' x {(x + 4)}^{\frac{1}{2}} = 2 y {(x + 4)}^{\frac{1}{2}} - y$

Step 6.5.2.2

Factor $2 y'$ out of $8 y'$ .

$2 y' x + 2 y' \cdot 4 - 2 y' x {(x + 4)}^{\frac{1}{2}} = 2 y {(x + 4)}^{\frac{1}{2}} - y$

Step 6.5.2.3

Factor $2 y'$ out of $- 2 y' x {(x + 4)}^{\frac{1}{2}}$ .

$2 y' x + 2 y' \cdot 4 + 2 y' (- 1 x {(x + 4)}^{\frac{1}{2}}) = 2 y {(x + 4)}^{\frac{1}{2}} - y$

Step 6.5.2.4

Factor $2 y'$ out of $2 y' x + 2 y' \cdot 4$ .

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

Step 6.5.2.5

Factor $2 y'$ out of $2 y' (x + 4) + 2 y' (- 1 x {(x + 4)}^{\frac{1}{2}})$ .

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

Step 6.5.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.5.4

Reorder terms.

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

Step 6.5.5

Divide each term in $2 y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4) = 2 y {(x + 4)}^{\frac{1}{2}} - y$ by $- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8$ and simplify.

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Step 6.5.5.1

Divide each term in $2 y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4) = 2 y {(x + 4)}^{\frac{1}{2}} - y$ by $- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8$ .

$\frac{2 y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4)}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2

Simplify the left side.

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Step 6.5.5.2.1

Factor $2$ out of $2 y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4)$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2

Cancel the common factors.

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Step 6.5.5.2.2.1

Factor $2$ out of $- 2 x {(x + 4)}^{\frac{1}{2}}$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 x + 8} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2.2

Factor $2$ out of $2 x$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 (x) + 8} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2.3

Factor $2$ out of $2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 (x)$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{2 (- x {(x + 4)}^{\frac{1}{2}} + x) + 8} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2.4

Factor $2$ out of $8$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{2 (- x {(x + 4)}^{\frac{1}{2}} + x) + 2 (4)} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2.5

Factor $2$ out of $2 (- x {(x + 4)}^{\frac{1}{2}} + x) + 2 (4)$ .

$\frac{2 (y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4))}{2 (- x {(x + 4)}^{\frac{1}{2}} + x + 4)} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.2.6

Cancel the common factor.

Step 6.5.5.2.2.7

Rewrite the expression.

$\frac{y' (- x {(x + 4)}^{\frac{1}{2}} + x + 4)}{- x {(x + 4)}^{\frac{1}{2}} + x + 4} = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.2.3

Cancel the common factor.

Step 6.5.5.2.4

Divide $y'$ by $1$ .

$y' = \frac{2 y {(x + 4)}^{\frac{1}{2}}}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8} + \frac{- y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.3

Simplify the right side.

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Step 6.5.5.3.1

Combine the numerators over the common denominator.

$y' = \frac{2 y {(x + 4)}^{\frac{1}{2}} - y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.3.2

Factor $y$ out of $2 y {(x + 4)}^{\frac{1}{2}} - y$ .

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Step 6.5.5.3.2.1

Factor $y$ out of $2 y {(x + 4)}^{\frac{1}{2}}$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}}) - y}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.3.2.2

Factor $y$ out of $- y$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}}) + y \cdot - 1}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.3.2.3

Factor $y$ out of $y (2 {(x + 4)}^{\frac{1}{2}}) + y \cdot - 1$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8}$

Step 6.5.5.3.3

Factor $2$ out of $- 2 x {(x + 4)}^{\frac{1}{2}} + 2 x + 8$ .

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Step 6.5.5.3.3.1

Factor $2$ out of $- 2 x {(x + 4)}^{\frac{1}{2}}$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 x + 8}$

Step 6.5.5.3.3.2

Factor $2$ out of $2 x$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 (x) + 8}$

Step 6.5.5.3.3.3

Factor $2$ out of $8$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 x + 2 \cdot 4}$

Step 6.5.5.3.3.4

Factor $2$ out of $2 (- x {(x + 4)}^{\frac{1}{2}}) + 2 x$ .

$y' = \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{2 (- x {(x + 4)}^{\frac{1}{2}} + x) + 2 \cdot 4}$

Step 6.5.5.3.3.5

Factor $2$ out of $2 (- x {(x + 4)}^{\frac{1}{2}} + x) + 2 \cdot 4$ .

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

Step 6.5.5.3.4

Factor $- 1$ out of $- x {(x + 4)}^{\frac{1}{2}}$ .

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

Step 6.5.5.3.5

Factor $- 1$ out of $x$ .

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

Step 6.5.5.3.6

Factor $- 1$ out of $- (x {(x + 4)}^{\frac{1}{2}}) - 1 (- x)$ .

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

Step 6.5.5.3.7

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 6.5.5.3.8

Factor $- 1$ out of $- (x {(x + 4)}^{\frac{1}{2}} - x) - 1 (- 4)$ .

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

Step 6.5.5.3.9

Rewrite negatives.

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Step 6.5.5.3.9.1

Rewrite $- (x {(x + 4)}^{\frac{1}{2}} - x - 4)$ as $- 1 (x {(x + 4)}^{\frac{1}{2}} - x - 4)$ .

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

Step 6.5.5.3.9.2

Move the negative in front of the fraction.

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

Step 7

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{y (2 {(x + 4)}^{\frac{1}{2}} - 1)}{2 (x {(x + 4)}^{\frac{1}{2}} - x - 4)}$