Calculus Examples

$f (x) = \ln (\sqrt{\frac{4 x + 4}{7 x - 4}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{4 x + 4}{7 x - 4}}$ as ${(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\ln ({(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}]$

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with ${(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} \frac{d}{d x} [{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}]$

Step 3

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 $g (x) = \frac{4 x + 4}{7 x - 4}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $\frac{4 x + 4}{7 x - 4}$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 3.2

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $\frac{4 x + 4}{7 x - 4}$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 4

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 5

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 7.2

Subtract $2$ from $1$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 8

Move the negative in front of the fraction.

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{4 x + 4}{7 x - 4}])$

Step 9

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 x + 4$ and $g (x) = 7 x - 4$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) \frac{d}{d x} [4 x + 4] - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10

Differentiate.

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Step 10.1

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) (\frac{d}{d x} [4 x] + \frac{d}{d x} [4]) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) (4 \frac{d}{d x} [x] + \frac{d}{d x} [4]) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.3

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) (4 \cdot 1 + \frac{d}{d x} [4]) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.4

Multiply $4$ by $1$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) (4 + \frac{d}{d x} [4]) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.5

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) (4 + 0) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.6

Simplify the expression.

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Step 10.6.1

Add $4$ and $0$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{(7 x - 4) \cdot 4 - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.6.2

Move $4$ to the left of $7 x - 4$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 \cdot (7 x - 4) - (4 x + 4) \frac{d}{d x} [7 x - 4]}{{(7 x - 4)}^{2}})$

Step 10.7

By the Sum Rule, the derivative of $7 x - 4$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 4]$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 4])}{{(7 x - 4)}^{2}})$

Step 10.8

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(7 x - 4)}^{2}})$

Step 10.9

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) (7 \cdot 1 + \frac{d}{d x} [- 4])}{{(7 x - 4)}^{2}})$

Step 10.10

Multiply $7$ by $1$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) (7 + \frac{d}{d x} [- 4])}{{(7 x - 4)}^{2}})$

Step 10.11

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

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) (7 + 0)}{{(7 x - 4)}^{2}})$

Step 10.12

Combine fractions.

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Step 10.12.1

Add $7$ and $0$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - (4 x + 4) \cdot 7}{{(7 x - 4)}^{2}})$

Step 10.12.2

Multiply $7$ by $- 1$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{1}{2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}} \frac{4 (7 x - 4) - 7 (4 x + 4)}{{(7 x - 4)}^{2}})$

Step 10.12.3

Multiply $\frac{4 (7 x - 4) - 7 (4 x + 4)}{{(7 x - 4)}^{2}}$ by $\frac{1}{2}$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{4 (7 x - 4) - 7 (4 x + 4)}{{(7 x - 4)}^{2} \cdot 2} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}})$

Step 10.12.4

Move $2$ to the left of ${(7 x - 4)}^{2}$ .

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{4 (7 x - 4) - 7 (4 x + 4)}{2 \cdot {(7 x - 4)}^{2}} {(\frac{4 x + 4}{7 x - 4})}^{- \frac{1}{2}})$

Step 11

Simplify.

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Step 11.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{1}{{(\frac{4 x + 4}{7 x - 4})}^{\frac{1}{2}}} (\frac{4 (7 x - 4) - 7 (4 x + 4)}{2 {(7 x - 4)}^{2}} {(\frac{7 x - 4}{4 x + 4})}^{\frac{1}{2}})$

Step 11.2

Apply the product rule to $\frac{4 x + 4}{7 x - 4}$ .

$\frac{1}{\frac{{(4 x + 4)}^{\frac{1}{2}}}{{(7 x - 4)}^{\frac{1}{2}}}} (\frac{4 (7 x - 4) - 7 (4 x + 4)}{2 {(7 x - 4)}^{2}} {(\frac{7 x - 4}{4 x + 4})}^{\frac{1}{2}})$

Step 11.3

Apply the product rule to $\frac{7 x - 4}{4 x + 4}$ .

$\frac{1}{\frac{{(4 x + 4)}^{\frac{1}{2}}}{{(7 x - 4)}^{\frac{1}{2}}}} (\frac{4 (7 x - 4) - 7 (4 x + 4)}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.4

Apply the distributive property.

$\frac{1}{\frac{{(4 x + 4)}^{\frac{1}{2}}}{{(7 x - 4)}^{\frac{1}{2}}}} (\frac{4 (7 x) + 4 \cdot - 4 - 7 (4 x + 4)}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.5

Apply the distributive property.

$\frac{1}{\frac{{(4 x + 4)}^{\frac{1}{2}}}{{(7 x - 4)}^{\frac{1}{2}}}} (\frac{4 (7 x) + 4 \cdot - 4 - 7 (4 x) - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6

Combine terms.

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Step 11.6.1

Multiply by the reciprocal of the fraction to divide by $\frac{{(4 x + 4)}^{\frac{1}{2}}}{{(7 x - 4)}^{\frac{1}{2}}}$ .

$1 \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{4 (7 x) + 4 \cdot - 4 - 7 (4 x) - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.2

Multiply $\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}}$ by $1$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{4 (7 x) + 4 \cdot - 4 - 7 (4 x) - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.3

Multiply $7$ by $4$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{28 x + 4 \cdot - 4 - 7 (4 x) - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.4

Multiply $4$ by $- 4$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{28 x - 16 - 7 (4 x) - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.5

Multiply $4$ by $- 7$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{28 x - 16 - 28 x - 7 \cdot 4}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.6

Multiply $- 7$ by $4$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{28 x - 16 - 28 x - 28}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.7

Subtract $28 x$ from $28 x$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{0 - 16 - 28}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.8

Subtract $16$ from $0$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{- 16 - 28}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.9

Subtract $28$ from $- 16$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{- 44}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.10

Cancel the common factor of $- 44$ and $2$ .

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Step 11.6.10.1

Factor $2$ out of $- 44$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{2 \cdot - 22}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.10.2

Cancel the common factors.

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Step 11.6.10.2.1

Factor $2$ out of $2 {(7 x - 4)}^{2}$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{2 \cdot - 22}{2 ({(7 x - 4)}^{2})} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.10.2.2

Cancel the common factor.

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{2 \cdot - 22}{2 {(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.10.2.3

Rewrite the expression.

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (\frac{- 22}{{(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.11

Move the negative in front of the fraction.

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(7 x - 4)}^{2}} \cdot \frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}})$

Step 11.6.12

Multiply $\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}}$ by $\frac{22}{{(7 x - 4)}^{2}}$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{2}})$

Step 11.6.13

Move $22$ to the left of ${(7 x - 4)}^{\frac{1}{2}}$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22 \cdot {(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{2}})$

Step 11.6.14

Move ${(7 x - 4)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{2} {(7 x - 4)}^{- \frac{1}{2}}})$

Step 11.6.15

Multiply ${(7 x - 4)}^{2}$ by ${(7 x - 4)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.6.15.1

Move ${(7 x - 4)}^{- \frac{1}{2}}$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} ({(7 x - 4)}^{- \frac{1}{2}} {(7 x - 4)}^{2})})$

Step 11.6.15.2

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

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{- \frac{1}{2} + 2}})$

Step 11.6.15.3

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

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}})$

Step 11.6.15.4

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

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}})$

Step 11.6.15.5

Combine the numerators over the common denominator.

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{- 1 + 2 \cdot 2}{2}}})$

Step 11.6.15.6

Simplify the numerator.

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Step 11.6.15.6.1

Multiply $2$ by $2$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{- 1 + 4}{2}}})$

Step 11.6.15.6.2

Add $- 1$ and $4$ .

$\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}} (- \frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}}})$

Step 11.6.16

Multiply $\frac{{(7 x - 4)}^{\frac{1}{2}}}{{(4 x + 4)}^{\frac{1}{2}}}$ by $\frac{22}{{(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}}}$ .

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{1}{2}} ({(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}})}$

Step 11.6.17

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

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Step 11.6.17.1

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

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{1}{2}} {(4 x + 4)}^{\frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.17.2

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

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{1}{2} + \frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.17.3

Combine the numerators over the common denominator.

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{1 + 1}{2}} {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.17.4

Add $1$ and $1$ .

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{\frac{2}{2}} {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.17.5

Divide $2$ by $2$ .

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{{(4 x + 4)}^{1} {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.18

Simplify ${(4 x + 4)}^{1} {(7 x - 4)}^{\frac{3}{2}}$ .

$- \frac{{(7 x - 4)}^{\frac{1}{2}} \cdot 22}{(4 x + 4) {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.19

Move $22$ to the left of ${(7 x - 4)}^{\frac{1}{2}}$ .

$- \frac{22 \cdot {(7 x - 4)}^{\frac{1}{2}}}{(4 x + 4) {(7 x - 4)}^{\frac{3}{2}}}$

Step 11.6.20

Move ${(7 x - 4)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{22}{(4 x + 4) {(7 x - 4)}^{\frac{3}{2}} {(7 x - 4)}^{- \frac{1}{2}}}$

Step 11.6.21

Simplify the denominator.

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Step 11.6.21.1

Multiply ${(7 x - 4)}^{\frac{3}{2}}$ by ${(7 x - 4)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.6.21.1.1

Move ${(7 x - 4)}^{- \frac{1}{2}}$ .

$- \frac{22}{(4 x + 4) ({(7 x - 4)}^{- \frac{1}{2}} {(7 x - 4)}^{\frac{3}{2}})}$

Step 11.6.21.1.2

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

$- \frac{22}{(4 x + 4) {(7 x - 4)}^{- \frac{1}{2} + \frac{3}{2}}}$

Step 11.6.21.1.3

Combine the numerators over the common denominator.

$- \frac{22}{(4 x + 4) {(7 x - 4)}^{\frac{- 1 + 3}{2}}}$

Step 11.6.21.1.4

Add $- 1$ and $3$ .

$- \frac{22}{(4 x + 4) {(7 x - 4)}^{\frac{2}{2}}}$

Step 11.6.21.1.5

Divide $2$ by $2$ .

$- \frac{22}{(4 x + 4) {(7 x - 4)}^{1}}$

Step 11.6.21.2

Simplify $(4 x + 4) {(7 x - 4)}^{1}$ .

$- \frac{22}{(4 x + 4) (7 x - 4)}$

Step 11.6.22

Cancel the common factor of $22$ and $4 x + 4$ .

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Step 11.6.22.1

Factor $2$ out of $22$ .

$- \frac{2 (11)}{(4 x + 4) (7 x - 4)}$

Step 11.6.22.2

Cancel the common factors.

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Step 11.6.22.2.1

Factor $2$ out of $(4 x + 4) (7 x - 4)$ .

$- \frac{2 (11)}{2 ((2 x + 2) (7 x - 4))}$

Step 11.6.22.2.2

Cancel the common factor.

$- \frac{2 \cdot 11}{2 ((2 x + 2) (7 x - 4))}$

Step 11.6.22.2.3

Rewrite the expression.

$- \frac{11}{(2 x + 2) (7 x - 4)}$

Step 11.7

Factor $2$ out of $2 x + 2$ .

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Step 11.7.1

Factor $2$ out of $2 x$ .

$- \frac{11}{(2 (x) + 2) (7 x - 4)}$

Step 11.7.2

Factor $2$ out of $2$ .

$- \frac{11}{(2 (x) + 2 (1)) (7 x - 4)}$

Step 11.7.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$- \frac{11}{2 (x + 1) (7 x - 4)}$