Calculus Examples

$f (x) = \ln (9 x - 7) \cdot \sqrt{7 x^{3} + 2 x^{2} - 5}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x^{3} + 2 x^{2} - 5}$ as ${(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\ln (9 x - 7) {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}]$

Step 2

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 $f (x) = \ln (9 x - 7)$ and $g (x) = {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ .

$\ln (9 x - 7) \frac{d}{d x} [{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}] + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

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 $g (x) = 7 x^{3} + 2 x^{2} - 5$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $7 x^{3} + 2 x^{2} - 5$ .

$\ln (9 x - 7) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 3.2

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

$\ln (9 x - 7) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 3.3

Replace all occurrences of $u_{1}$ with $7 x^{3} + 2 x^{2} - 5$ .

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 4

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

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 5

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

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 6

Combine the numerators over the common denominator.

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 7.2

Subtract $2$ from $1$ .

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{\frac{- 1}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8

Differentiate.

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Step 8.1

Move the negative in front of the fraction.

$\ln (9 x - 7) (\frac{1}{2} {(7 x^{3} + 2 x^{2} - 5)}^{- \frac{1}{2}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.2

Combine fractions.

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Step 8.2.1

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

$\ln (9 x - 7) (\frac{{(7 x^{3} + 2 x^{2} - 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.2.2

Move ${(7 x^{3} + 2 x^{2} - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\ln (9 x - 7) (\frac{1}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.2.3

Combine $\frac{1}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}$ and $\ln (9 x - 7)$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} \frac{d}{d x} [7 x^{3} + 2 x^{2} - 5] + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.3

By the Sum Rule, the derivative of $7 x^{3} + 2 x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5]$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.4

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

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.5

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

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (7 (3 x^{2}) + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.6

Multiply $3$ by $7$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.7

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

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.8

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

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 2 (2 x) + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.9

Multiply $2$ by $2$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x + \frac{d}{d x} [- 5]) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.10

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x + 0) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 8.11

Add $21 x^{2} + 4 x$ and $0$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \frac{d}{d x} [\ln (9 x - 7)]$

Step 9

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 $g (x) = 9 x - 7$ .

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Step 9.1

To apply the Chain Rule, set $u_{2}$ as $9 x - 7$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [9 x - 7])$

Step 9.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} (\frac{1}{u_{2}} \frac{d}{d x} [9 x - 7])$

Step 9.3

Replace all occurrences of $u_{2}$ with $9 x - 7$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} (\frac{1}{9 x - 7} \frac{d}{d x} [9 x - 7])$

Step 10

Differentiate.

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Step 10.1

Combine $\frac{1}{9 x - 7}$ and ${(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} \frac{d}{d x} [9 x - 7]$

Step 10.2

By the Sum Rule, the derivative of $9 x - 7$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [- 7]$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} (\frac{d}{d x} [9 x] + \frac{d}{d x} [- 7])$

Step 10.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} (9 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 10.4

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

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} (9 \cdot 1 + \frac{d}{d x} [- 7])$

Step 10.5

Multiply $9$ by $1$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} (9 + \frac{d}{d x} [- 7])$

Step 10.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} (9 + 0)$

Step 10.7

Combine fractions.

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Step 10.7.1

Add $9$ and $0$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7} \cdot 9$

Step 10.7.2

Combine $\frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$ and $9$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{{(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} \cdot 9}{9 x - 7}$

Step 10.7.3

Move $9$ to the left of ${(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) + \frac{9 \cdot {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$

Step 11

To write $\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x)$ as a fraction with a common denominator, multiply by $\frac{9 x - 7}{9 x - 7}$ .

$\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) \cdot \frac{9 x - 7}{9 x - 7} + \frac{9 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$

Step 12

Combine $\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x)$ and $\frac{9 x - 7}{9 x - 7}$ .

$\frac{\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) (9 x - 7)}{9 x - 7} + \frac{9 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$

Step 13

Combine the numerators over the common denominator.

$\frac{\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} (21 x^{2} + 4 x) (9 x - 7) + 9 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$

Step 14

Simplify.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Add parentheses.

$\frac{\frac{\ln (9 x - 7)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} ((21 x^{2} + 4 x) (9 x - 7)) + 9 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}{9 x - 7}$

Step 14.1.2

Let $u_{3} = {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ . Substitute $u_{3}$ for all occurrences of ${(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}$ .

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 {u_{3}}^{2}}{2 u_{3}}}{9 x - 7}$

Step 14.1.3

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

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 {({(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}})}^{2}}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4

Simplify each term.

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Step 14.1.4.1

Multiply the exponents in ${({(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 14.1.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2} \cdot 2}}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.1.2

Cancel the common factor of $2$ .

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Step 14.1.4.1.2.1

Cancel the common factor.

Step 14.1.4.1.2.2

Rewrite the expression.

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 {(7 x^{3} + 2 x^{2} - 5)}^{1}}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.2

Simplify.

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 (7 x^{3} + 2 x^{2} - 5)}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.3

Apply the distributive property.

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 18 (7 x^{3}) + 18 (2 x^{2}) + 18 \cdot - 5}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.4

Simplify.

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Step 14.1.4.4.1

Multiply $7$ by $18$ .

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 18 (2 x^{2}) + 18 \cdot - 5}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.4.2

Multiply $2$ by $18$ .

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 36 x^{2} + 18 \cdot - 5}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.1.4.4.3

Multiply $18$ by $- 5$ .

$\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 36 x^{2} - 90}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$

Step 14.2

Combine terms.

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Step 14.2.1

Rewrite $\frac{\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 36 x^{2} - 90}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}}{9 x - 7}$ as a product.

$\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 36 x^{2} - 90}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}} \cdot \frac{1}{9 x - 7}$

Step 14.2.2

Multiply $\frac{x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) + 126 x^{3} + 36 x^{2} - 90}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}}}$ by $\frac{1}{9 x - 7}$ .

Step 14.3

Reorder terms.

$\frac{126 x^{3} + 36 x^{2} + x^{3} \ln ({(9 x - 7)}^{189}) - x^{2} \ln ({(9 x - 7)}^{147}) + x^{2} \ln ({(9 x - 7)}^{36}) - x \ln ({(9 x - 7)}^{28}) - 90}{2 {(7 x^{3} + 2 x^{2} - 5)}^{\frac{1}{2}} (9 x - 7)}$