Calculus Examples

$y = \log_{9} (\frac{x^{2}}{10 \sqrt{x + 1}})$

Step 1

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

$y = \log_{9} (\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\log_{9} (\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{9} (x)$ and $g (x) = \frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}$ .

$\frac{d}{d u_{1}} [\log_{9} (u_{1})] \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.1.2

The derivative of $\log_{9} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (9)}$ .

$\frac{1}{u_{1} \ln (9)} \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}$ .

$\frac{1}{\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}} \ln (9)} \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.2

Combine $\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}$ and $\ln (9)$ .

$\frac{1}{\frac{x^{2} \ln (9)}{10 {(x + 1)}^{\frac{1}{2}}}} \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.3

Multiply by the reciprocal of the fraction to divide by $\frac{x^{2} \ln (9)}{10 {(x + 1)}^{\frac{1}{2}}}$ .

$1 \frac{10 {(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.4

Differentiate using the Constant Multiple Rule.

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Step 4.4.1

Multiply $\frac{10 {(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)}$ by $1$ .

$\frac{10 {(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \frac{d}{d x} [\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}]$

Step 4.4.2

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{10 {(x + 1)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [\frac{x^{2}}{{(x + 1)}^{\frac{1}{2}}}]$ .

$\frac{10 {(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} (\frac{1}{10} \frac{d}{d x} [\frac{x^{2}}{{(x + 1)}^{\frac{1}{2}}}])$

Step 4.4.3

Simplify terms.

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Step 4.4.3.1

Multiply $\frac{1}{10}$ by $\frac{10 {(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)}$ .

$\frac{10 {(x + 1)}^{\frac{1}{2}}}{10 (x^{2} \ln (9))} \frac{d}{d x} [\frac{x^{2}}{{(x + 1)}^{\frac{1}{2}}}]$

Step 4.4.3.2

Cancel the common factor.

$\frac{10 {(x + 1)}^{\frac{1}{2}}}{10 x^{2} \ln (9)} \frac{d}{d x} [\frac{x^{2}}{{(x + 1)}^{\frac{1}{2}}}]$

Step 4.4.3.3

Rewrite the expression.

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

Step 4.5

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 $f (x) = x^{2}$ and $g (x) = {(x + 1)}^{\frac{1}{2}}$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Cancel the common factor of $2$ .

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Step 4.6.2.1

Cancel the common factor.

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

Step 4.6.2.2

Rewrite the expression.

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

Step 4.7

Simplify.

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

Step 4.8

Differentiate using the Power Rule.

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Step 4.8.1

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

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

Step 4.8.2

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

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

Step 4.9

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

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Step 4.9.1

To apply the Chain Rule, set $u_{2}$ as $x + 1$ .

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

Step 4.9.2

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

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

Step 4.9.3

Replace all occurrences of $u_{2}$ with $x + 1$ .

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Combine the numerators over the common denominator.

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

Step 4.13

Simplify the numerator.

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Step 4.13.1

Multiply $- 1$ by $2$ .

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

Step 4.13.2

Subtract $2$ from $1$ .

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

Step 4.14

Combine fractions.

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Step 4.14.1

Move the negative in front of the fraction.

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

Step 4.14.2

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

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

Step 4.14.3

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

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

Step 4.14.4

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

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

Step 4.15

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 4.16

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

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

Step 4.17

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{2 {(x + 1)}^{\frac{1}{2}} x - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}} (1 + 0)}{x + 1}$

Step 4.18

Simplify the expression.

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Step 4.18.1

Add $1$ and $0$ .

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{2 {(x + 1)}^{\frac{1}{2}} x - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}} \cdot 1}{x + 1}$

Step 4.18.2

Multiply $- 1$ by $1$ .

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{2 {(x + 1)}^{\frac{1}{2}} x - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}}{x + 1}$

Step 4.19

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

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Step 4.19.1

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

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{2 x {(x + 1)}^{\frac{1}{2}} - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}}{x + 1}$

Step 4.19.2

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

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{2 x {(x + 1)}^{\frac{1}{2}} \cdot \frac{2 {(x + 1)}^{\frac{1}{2}}}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}}{x + 1}$

Step 4.19.3

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

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{\frac{2 x {(x + 1)}^{\frac{1}{2}} (2 {(x + 1)}^{\frac{1}{2}})}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}}{x + 1}$

Step 4.19.4

Combine the numerators over the common denominator.

$\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)} \cdot \frac{\frac{2 x {(x + 1)}^{\frac{1}{2}} (2 {(x + 1)}^{\frac{1}{2}}) - x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}}{x + 1}$

Step 4.20

Multiply $2$ by $2$ .

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

Step 4.21

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

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Step 4.21.1

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

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

Step 4.21.2

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

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

Step 4.21.3

Combine the numerators over the common denominator.

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

Step 4.21.4

Add $1$ and $1$ .

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

Step 4.21.5

Divide $2$ by $2$ .

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

Step 4.22

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

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

Step 4.23

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

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

Step 4.24

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

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

Step 4.25

Raise $x + 1$ to the power of $1$ .

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

Step 4.26

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

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

Step 4.27

Simplify the expression.

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Step 4.27.1

Write $1$ as a fraction with a common denominator.

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

Step 4.27.2

Combine the numerators over the common denominator.

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

Step 4.27.3

Add $2$ and $1$ .

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

Step 4.28

Multiply $\frac{{(x + 1)}^{\frac{1}{2}}}{x^{2} \ln (9)}$ by $\frac{4 x (x + 1) - x^{2}}{2 {(x + 1)}^{\frac{3}{2}}}$ .

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

Step 4.29

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

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

Step 4.30

Simplify the denominator.

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Step 4.30.1

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

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Step 4.30.1.1

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

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

Step 4.30.1.2

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

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

Step 4.30.1.3

Combine the numerators over the common denominator.

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

Step 4.30.1.4

Add $- 1$ and $3$ .

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

Step 4.30.1.5

Divide $2$ by $2$ .

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

Step 4.30.2

Simplify $x^{2} \ln (9) (2 {(x + 1)}^{1})$ .

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

Step 4.31

Simplify.

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Step 4.31.1

Apply the distributive property.

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

Step 4.31.2

Apply the distributive property.

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

Step 4.31.3

Apply the distributive property.

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

Step 4.31.4

Simplify the numerator.

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Step 4.31.4.1

Simplify each term.

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Step 4.31.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.4.1.1.1

Move $x$ .

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

Step 4.31.4.1.1.2

Multiply $x$ by $x$ .

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

Step 4.31.4.1.2

Multiply $4$ by $1$ .

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

Step 4.31.4.2

Subtract $x^{2}$ from $4 x^{2}$ .

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

Step 4.31.5

Combine terms.

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Step 4.31.5.1

Raise $x$ to the power of $1$ .

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

Step 4.31.5.2

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

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

Step 4.31.5.3

Add $1$ and $2$ .

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

Step 4.31.5.4

Move $2$ to the left of $x^{3} \ln (9)$ .

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

Step 4.31.5.5

Multiply $2$ by $1$ .

$\frac{3 x^{2} + 4 x}{2 x^{3} \ln (9) + x^{2} \ln (9) \cdot 2}$

Step 4.31.5.6

Move $2$ to the left of $x^{2} \ln (9)$ .

$\frac{3 x^{2} + 4 x}{2 x^{3} \ln (9) + 2 x^{2} \ln (9)}$

Step 4.31.6

Factor $x$ out of $3 x^{2} + 4 x$ .

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Step 4.31.6.1

Factor $x$ out of $3 x^{2}$ .

$\frac{x (3 x) + 4 x}{2 x^{3} \ln (9) + 2 x^{2} \ln (9)}$

Step 4.31.6.2

Factor $x$ out of $4 x$ .

$\frac{x (3 x) + x \cdot 4}{2 x^{3} \ln (9) + 2 x^{2} \ln (9)}$

Step 4.31.6.3

Factor $x$ out of $x (3 x) + x \cdot 4$ .

$\frac{x (3 x + 4)}{2 x^{3} \ln (9) + 2 x^{2} \ln (9)}$

Step 4.31.7

Factor $2 x^{2} \ln (9)$ out of $2 x^{3} \ln (9) + 2 x^{2} \ln (9)$ .

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Step 4.31.7.1

Factor $2 x^{2} \ln (9)$ out of $2 x^{3} \ln (9)$ .

$\frac{x (3 x + 4)}{2 x^{2} \ln (9) (x) + 2 x^{2} \ln (9)}$

Step 4.31.7.2

Factor $2 x^{2} \ln (9)$ out of $2 x^{2} \ln (9)$ .

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

Step 4.31.7.3

Factor $2 x^{2} \ln (9)$ out of $2 x^{2} \ln (9) (x) + 2 x^{2} \ln (9) (1)$ .

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

Step 4.31.8

Cancel the common factors.

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Step 4.31.8.1

Factor $x$ out of $2 x^{2} \ln (9) (x + 1)$ .

$\frac{x (3 x + 4)}{x (2 x \ln (9) (x + 1))}$

Step 4.31.8.2

Cancel the common factor.

$\frac{x (3 x + 4)}{x (2 x \ln (9) (x + 1))}$

Step 4.31.8.3

Rewrite the expression.

$\frac{3 x + 4}{2 x \ln (9) (x + 1)}$

Step 5

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

$y' = \frac{3 x + 4}{2 x \ln (9) (x + 1)}$

Step 6

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

$\frac{d y}{d x} = \frac{3 x + 4}{2 x \ln (9) (x + 1)}$