Calculus Examples

$y = \log_{5} (\sqrt{{(\frac{7 x}{3 x + 2})}^{\ln (5)}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(\frac{7 x}{3 x + 2})}^{\ln (5)}}$ as ${(\frac{7 x}{3 x + 2})}^{\frac{\ln (5)}{2}}$ .

$\frac{d}{d x} [\log_{5} ({(\frac{7 x}{3 x + 2})}^{\frac{\ln (5)}{2}})]$

Step 2

Rewrite $\frac{\ln (5)}{2}$ as $\frac{1}{2} \ln (5)$ .

$\frac{d}{d x} [\log_{5} ({(\frac{7 x}{3 x + 2})}^{\frac{1}{2} \ln (5)})]$

Step 3

Simplify $\frac{1}{2} \ln (5)$ by moving $\frac{1}{2}$ inside the logarithm.

$\frac{d}{d x} [\log_{5} ({(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})})]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{5} (x)$ and $g (x) = {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}$ .

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Step 4.1

To apply the Chain Rule, set $u_{1}$ as ${(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}$ .

$\frac{d}{d u_{1}} [\log_{5} (u_{1})] \frac{d}{d x} [{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}]$

Step 4.2

The derivative of $\log_{5} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (5)}$ .

$\frac{1}{u_{1} \ln (5)} \frac{d}{d x} [{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}]$

Step 4.3

Replace all occurrences of $u_{1}$ with ${(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} \frac{d}{d x} [{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})}]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\ln (5^{\frac{1}{2}})}$ and $g (x) = \frac{7 x}{3 x + 2}$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $\frac{7 x}{3 x + 2}$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{d}{d u_{2}} [{u_{2}}^{\ln (5^{\frac{1}{2}})}] \frac{d}{d x} [\frac{7 x}{3 x + 2}])$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \ln (5^{\frac{1}{2}})$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {u_{2}}^{\ln (5^{\frac{1}{2}}) - 1} \frac{d}{d x} [\frac{7 x}{3 x + 2}])$

Step 5.3

Replace all occurrences of $u_{2}$ with $\frac{7 x}{3 x + 2}$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} \frac{d}{d x} [\frac{7 x}{3 x + 2}])$

Step 6

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7 x}{3 x + 2}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{x}{3 x + 2}]$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{d}{d x} [\frac{x}{3 x + 2}]))$

Step 7

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 $f (x) = x$ and $g (x) = 3 x + 2$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{(3 x + 2) \frac{d}{d x} [x] - x \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}))$

Step 8

Differentiate.

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Step 8.1

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

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{(3 x + 2) \cdot 1 - x \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}))$

Step 8.2

Multiply $3 x + 2$ by $1$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}))$

Step 8.3

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x (\frac{d}{d x} [3 x] + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}))$

Step 8.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x (3 \frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}))$

Step 8.5

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

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x (3 \cdot 1 + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}))$

Step 8.6

Multiply $3$ by $1$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x (3 + \frac{d}{d x} [2])}{{(3 x + 2)}^{2}}))$

Step 8.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x (3 + 0)}{{(3 x + 2)}^{2}}))$

Step 8.8

Simplify terms.

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Step 8.8.1

Add $3$ and $0$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - x \cdot 3}{{(3 x + 2)}^{2}}))$

Step 8.8.2

Multiply $3$ by $- 1$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{3 x + 2 - 3 x}{{(3 x + 2)}^{2}}))$

Step 8.8.3

Subtract $3 x$ from $3 x$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{0 + 2}{{(3 x + 2)}^{2}}))$

Step 8.8.4

Add $0$ and $2$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} (7 \frac{2}{{(3 x + 2)}^{2}}))$

Step 8.8.5

Combine $7$ and $\frac{2}{{(3 x + 2)}^{2}}$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} \frac{7 \cdot 2}{{(3 x + 2)}^{2}})$

Step 8.8.6

Multiply $7$ by $2$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\ln (5^{\frac{1}{2}}) {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1} \frac{14}{{(3 x + 2)}^{2}})$

Step 8.8.7

Combine $\frac{14}{{(3 x + 2)}^{2}}$ and $\ln (5^{\frac{1}{2}})$ .

$\frac{1}{{(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1})$

Step 9

Simplify.

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Step 9.1

Apply the product rule to $\frac{7 x}{3 x + 2}$ .

$\frac{1}{\frac{{(7 x)}^{\ln (5^{\frac{1}{2}})}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1})$

Step 9.2

Apply the product rule to $7 x$ .

$\frac{1}{\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} {(\frac{7 x}{3 x + 2})}^{\ln (5^{\frac{1}{2}}) - 1})$

Step 9.3

Apply the product rule to $\frac{7 x}{3 x + 2}$ .

$\frac{1}{\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{{(7 x)}^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}})$

Step 9.4

Apply the product rule to $7 x$ .

$\frac{1}{\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{7^{\ln (5^{\frac{1}{2}}) - 1} x^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}})$

Step 9.5

Combine terms.

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Step 9.5.1

Combine $\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}$ and $\ln (5)$ .

$\frac{1}{\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{7^{\ln (5^{\frac{1}{2}}) - 1} x^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}})$

Step 9.5.2

Multiply by the reciprocal of the fraction to divide by $\frac{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}$ .

$1 \frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{7^{\ln (5^{\frac{1}{2}}) - 1} x^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}})$

Step 9.5.3

Multiply $\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)}$ by $1$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{7^{\ln (5^{\frac{1}{2}}) - 1} x^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}})$

Step 9.5.4

Move $7^{\ln (5^{\frac{1}{2}}) - 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{x^{\ln (5^{\frac{1}{2}}) - 1}}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)}})$

Step 9.5.5

Move $x^{\ln (5^{\frac{1}{2}}) - 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{1}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}})$

Step 9.5.6

Move ${(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} (\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{- (\ln (5^{\frac{1}{2}}) - 1)}}{7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}})$

Step 9.5.7

Multiply $\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2}}$ by $\frac{{(3 x + 2)}^{- (\ln (5^{\frac{1}{2}}) - 1)}}{7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} \cdot \frac{14 \ln (5^{\frac{1}{2}}) {(3 x + 2)}^{- (\ln (5^{\frac{1}{2}}) - 1)}}{{(3 x + 2)}^{2} (7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)})}$

Step 9.5.8

Move ${(3 x + 2)}^{- (\ln (5^{\frac{1}{2}}) - 1)}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} \cdot \frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{2} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)} {(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}}$

Step 9.5.9

Multiply ${(3 x + 2)}^{2}$ by ${(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}$ by adding the exponents.

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Step 9.5.9.1

Move ${(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} \cdot \frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1} {(3 x + 2)}^{2} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}}$

Step 9.5.9.2

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

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} \cdot \frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) - 1 + 2} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}}$

Step 9.5.9.3

Add $- 1$ and $2$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)} \cdot \frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}}$

Step 9.5.10

Multiply $\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})}}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5)}$ by $\frac{14 \ln (5^{\frac{1}{2}})}{{(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)}}$ .

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})} (14 \ln (5^{\frac{1}{2}}))}{7^{\ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5) ({(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1} \cdot 7^{- (\ln (5^{\frac{1}{2}}) - 1)} x^{- (\ln (5^{\frac{1}{2}}) - 1)})}$

Step 9.5.11

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

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})} (14 \ln (5^{\frac{1}{2}}))}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{\ln (5^{\frac{1}{2}})} \ln (5) ({(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1} x^{- (\ln (5^{\frac{1}{2}}) - 1)})}$

Step 9.5.12

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

$\frac{{(3 x + 2)}^{\ln (5^{\frac{1}{2}})} (14 \ln (5^{\frac{1}{2}}))}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1}}$

Step 9.5.13

Move $14$ to the left of ${(3 x + 2)}^{\ln (5^{\frac{1}{2}})}$ .

$\frac{14 \cdot {(3 x + 2)}^{\ln (5^{\frac{1}{2}})} \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1}}$

Step 9.5.14

Move ${(3 x + 2)}^{\ln (5^{\frac{1}{2}})}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1} {(3 x + 2)}^{- \ln (5^{\frac{1}{2}})}}$

Step 9.5.15

Simplify the denominator.

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Step 9.5.15.1

Multiply ${(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1}$ by ${(3 x + 2)}^{- \ln (5^{\frac{1}{2}})}$ by adding the exponents.

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Step 9.5.15.1.1

Move ${(3 x + 2)}^{- \ln (5^{\frac{1}{2}})}$ .

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) ({(3 x + 2)}^{- \ln (5^{\frac{1}{2}})} {(3 x + 2)}^{\ln (5^{\frac{1}{2}}) + 1})}$

Step 9.5.15.1.2

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

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{- \ln (5^{\frac{1}{2}}) + \ln (5^{\frac{1}{2}}) + 1}}$

Step 9.5.15.1.3

Add $- \ln (5^{\frac{1}{2}})$ and $\ln (5^{\frac{1}{2}})$ .

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{0 + 1}}$

Step 9.5.15.1.4

Add $0$ and $1$ .

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{1}}$

Step 9.5.15.2

Simplify $7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) {(3 x + 2)}^{1}$ .

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} \ln (5) (3 x + 2)}$

Step 9.6

Reorder terms.

$\frac{14 \ln (5^{\frac{1}{2}})}{7^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} x^{- (\ln (5^{\frac{1}{2}}) - 1) + \ln (5^{\frac{1}{2}})} (3 x + 2) \ln (5)}$

Step 9.7