Calculus Examples

$y = 9 x^{5} \arcsin (9 x^{5}) + \sqrt{1 - 81 x^{10}}$

Step 1

By the Sum Rule, the derivative of $9 x^{5} \arcsin (9 x^{5}) + \sqrt{1 - 81 x^{10}}$ with respect to $x$ is $\frac{d}{d x} [9 x^{5} \arcsin (9 x^{5})] + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$ .

$\frac{d}{d x} [9 x^{5} \arcsin (9 x^{5})] + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2

Evaluate $\frac{d}{d x} [9 x^{5} \arcsin (9 x^{5})]$ .

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Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5} \arcsin (9 x^{5})$ with respect to $x$ is $9 \frac{d}{d x} [x^{5} \arcsin (9 x^{5})]$ .

$9 \frac{d}{d x} [x^{5} \arcsin (9 x^{5})] + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.2

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 $f (x) = x^{5}$ and $g (x) = \arcsin (9 x^{5})$ .

$9 (x^{5} \frac{d}{d x} [\arcsin (9 x^{5})] + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = 9 x^{5}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $9 x^{5}$ .

$9 (x^{5} (\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [9 x^{5}]) + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.3.2

The derivative of $\arcsin (u_{1})$ with respect to $u_{1}$ is $\frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [9 x^{5}]) + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $9 x^{5}$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - {(9 x^{5})}^{2}}} \frac{d}{d x} [9 x^{5}]) + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.4

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - {(9 x^{5})}^{2}}} (9 \frac{d}{d x} [x^{5}])) + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.5

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

$9 (x^{5} (\frac{1}{\sqrt{1 - {(9 x^{5})}^{2}}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.6

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

$9 (x^{5} (\frac{1}{\sqrt{1 - {(9 x^{5})}^{2}}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.7

Factor $9$ out of $9 x^{5}$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - {(9 (x^{5}))}^{2}}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.8

Apply the product rule to $9 (x^{5})$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - (9^{2} {(x^{5})}^{2})}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.9

Raise $9$ to the power of $2$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - (81 {(x^{5})}^{2})}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.10

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.10.1

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

$9 (x^{5} (\frac{1}{\sqrt{1 - (81 x^{5 \cdot 2})}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.10.2

Multiply $5$ by $2$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - (81 x^{10})}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.11

Multiply $81$ by $- 1$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - 81 x^{10}}} (9 (5 x^{4}))) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.12

Multiply $5$ by $9$ .

$9 (x^{5} (\frac{1}{\sqrt{1 - 81 x^{10}}} (45 x^{4})) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.13

Combine $45$ and $\frac{1}{\sqrt{1 - 81 x^{10}}}$ .

$9 (x^{5} (\frac{45}{\sqrt{1 - 81 x^{10}}} x^{4}) + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.14

Combine $\frac{45}{\sqrt{1 - 81 x^{10}}}$ and $x^{4}$ .

$9 (x^{5} \frac{45 x^{4}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.15

Combine $x^{5}$ and $\frac{45 x^{4}}{\sqrt{1 - 81 x^{10}}}$ .

$9 (\frac{x^{5} (45 x^{4})}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.16

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 2.16.1

Move $x^{4}$ .

$9 (\frac{x^{4} x^{5} \cdot 45}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.16.2

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

$9 (\frac{x^{4 + 5} \cdot 45}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.16.3

Add $4$ and $5$ .

$9 (\frac{x^{9} \cdot 45}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 2.17

Move $45$ to the left of $x^{9}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$

Step 3

Evaluate $\frac{d}{d x} [\sqrt{1 - 81 x^{10}}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 81 x^{10}}$ as ${(1 - 81 x^{10})}^{\frac{1}{2}}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d x} [{(1 - 81 x^{10})}^{\frac{1}{2}}]$

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 $g (x) = 1 - 81 x^{10}$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $1 - 81 x^{10}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [1 - 81 x^{10}]$

Step 3.2.2

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

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - 81 x^{10}]$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $1 - 81 x^{10}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - 81 x^{10}]$

Step 3.3

By the Sum Rule, the derivative of $1 - 81 x^{10}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 81 x^{10}]$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1} (\frac{d}{d x} [1] + \frac{d}{d x} [- 81 x^{10}])$

Step 3.4

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

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- 81 x^{10}])$

Step 3.5

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81 x^{10}$ with respect to $x$ is $- 81 \frac{d}{d x} [x^{10}]$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1} (0 - 81 \frac{d}{d x} [x^{10}])$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1} (0 - 81 (10 x^{9}))$

Step 3.7

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

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - 81 (10 x^{9}))$

Step 3.8

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

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - 81 (10 x^{9}))$

Step 3.9

Combine the numerators over the common denominator.

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1 - 1 \cdot 2}{2}} (0 - 81 (10 x^{9}))$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{1 - 2}{2}} (0 - 81 (10 x^{9}))$

Step 3.10.2

Subtract $2$ from $1$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{\frac{- 1}{2}} (0 - 81 (10 x^{9}))$

Step 3.11

Move the negative in front of the fraction.

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{- \frac{1}{2}} (0 - 81 (10 x^{9}))$

Step 3.12

Multiply $10$ by $- 81$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{- \frac{1}{2}} (0 - 810 x^{9})$

Step 3.13

Subtract $810 x^{9}$ from $0$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{1}{2} {(1 - 81 x^{10})}^{- \frac{1}{2}} (- 810 x^{9})$

Step 3.14

Combine $\frac{1}{2}$ and ${(1 - 81 x^{10})}^{- \frac{1}{2}}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{{(1 - 81 x^{10})}^{- \frac{1}{2}}}{2} (- 810 x^{9})$

Step 3.15

Combine $- 810$ and $\frac{{(1 - 81 x^{10})}^{- \frac{1}{2}}}{2}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{- 810 {(1 - 81 x^{10})}^{- \frac{1}{2}}}{2} x^{9}$

Step 3.16

Combine $\frac{- 810 {(1 - 81 x^{10})}^{- \frac{1}{2}}}{2}$ and $x^{9}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{- 810 {(1 - 81 x^{10})}^{- \frac{1}{2}} x^{9}}{2}$

Step 3.17

Move ${(1 - 81 x^{10})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{- 810 x^{9}}{2 {(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 3.18

Factor $2$ out of $- 810 x^{9}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{2 (- 405 x^{9})}{2 {(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 3.19

Cancel the common factors.

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Step 3.19.1

Factor $2$ out of $2 {(1 - 81 x^{10})}^{\frac{1}{2}}$ .

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{2 (- 405 x^{9})}{2 ({(1 - 81 x^{10})}^{\frac{1}{2}})}$

Step 3.19.2

Cancel the common factor.

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{2 (- 405 x^{9})}{2 {(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 3.19.3

Rewrite the expression.

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) + \frac{- 405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 3.20

Move the negative in front of the fraction.

$9 (\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + \arcsin (9 x^{5}) (5 x^{4})) - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$9 \frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}} + 9 (\arcsin (9 x^{5}) (5 x^{4})) - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $9$ and $\frac{45 x^{9}}{\sqrt{1 - 81 x^{10}}}$ .

$\frac{9 (45 x^{9})}{\sqrt{1 - 81 x^{10}}} + 9 (\arcsin (9 x^{5}) (5 x^{4})) - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.2.2

Multiply $45$ by $9$ .

$\frac{405 x^{9}}{\sqrt{1 - 81 x^{10}}} + 9 (\arcsin (9 x^{5}) (5 x^{4})) - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.2.3

Multiply $5$ by $9$ .

$\frac{405 x^{9}}{\sqrt{1 - 81 x^{10}}} + 45 \arcsin (9 x^{5}) x^{4} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.3

Reorder terms.

$\frac{405 x^{9}}{\sqrt{1 - 81 x^{10}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4

Simplify each term.

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Step 4.4.1

Simplify the denominator.

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Step 4.4.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{405 x^{9}}{\sqrt{1^{2} - 81 x^{10}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.1.2

Rewrite $81 x^{10}$ as ${(9 x^{5})}^{2}$ .

$\frac{405 x^{9}}{\sqrt{1^{2} - {(9 x^{5})}^{2}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 9 x^{5}$ .

$\frac{405 x^{9}}{\sqrt{(1 + 9 x^{5}) (1 - (9 x^{5}))}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.1.4

Multiply $9$ by $- 1$ .

$\frac{405 x^{9}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.2

Multiply $\frac{405 x^{9}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}$ by $\frac{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}$ .

$\frac{405 x^{9}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}} \cdot \frac{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

Multiply $\frac{405 x^{9}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}$ by $\frac{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.2

Raise $\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}$ to the power of $1$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{1} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.3

Raise $\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}$ to the power of $1$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{1} {\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{1}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.4

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

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{1 + 1}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.5

Add $1$ and $1$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{2}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6

Rewrite ${\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}^{2}$ as $(1 + 9 x^{5}) (1 - 9 x^{5})$ .

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Step 4.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}$ as ${((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{1}{2}}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{({((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{1}{2}})}^{2}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6.2

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

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{1}{2} \cdot 2}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6.3

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

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{2}{2}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.4.1

Cancel the common factor.

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{2}{2}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6.4.2

Rewrite the expression.

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{{((1 + 9 x^{5}) (1 - 9 x^{5}))}^{1}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.4.3.6.5

Simplify.

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{(1 + 9 x^{5}) (1 - 9 x^{5})} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.5

To write $\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$ as a fraction with a common denominator, multiply by $\frac{{(1 - 81 x^{10})}^{\frac{1}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{(1 + 9 x^{5}) (1 - 9 x^{5})} \cdot \frac{{(1 - 81 x^{10})}^{\frac{1}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.6

To write $- \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{(1 + 9 x^{5}) (1 - 9 x^{5})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$ .

Step 4.7

Write each expression with a common denominator of $(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.7.1

Multiply $\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$ by $\frac{{(1 - 81 x^{10})}^{\frac{1}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}} - \frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}} \cdot \frac{(1 + 9 x^{5}) (1 - 9 x^{5})}{(1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.7.2

Multiply $\frac{405 x^{9}}{{(1 - 81 x^{10})}^{\frac{1}{2}}}$ by $\frac{(1 + 9 x^{5}) (1 - 9 x^{5})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}} - \frac{405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} ((1 + 9 x^{5}) (1 - 9 x^{5}))} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.7.3

Reorder the factors of $(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} - \frac{405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} ((1 + 9 x^{5}) (1 - 9 x^{5}))} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.7.4

Reorder the factors of ${(1 - 81 x^{10})}^{\frac{1}{2}} ((1 + 9 x^{5}) (1 - 9 x^{5}))$ .

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} - \frac{405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.8

Combine the numerators over the common denominator.

$\frac{405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Factor $405 x^{9}$ out of $405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))$ .

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Step 4.9.1.1

Factor $405 x^{9}$ out of $405 x^{9} \sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}) - 405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.1.2

Factor $405 x^{9}$ out of $- 405 x^{9} ((1 + 9 x^{5}) (1 - 9 x^{5}))$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}) + 405 x^{9} (- ((1 + 9 x^{5}) (1 - 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.1.3

Factor $405 x^{9}$ out of $405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}}) + 405 x^{9} (- ((1 + 9 x^{5}) (1 - 9 x^{5})))$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - ((1 + 9 x^{5}) (1 - 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.2

Expand $(1 + 9 x^{5}) (1 - 9 x^{5})$ using the FOIL Method.

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Step 4.9.2.1

Apply the distributive property.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 (1 - 9 x^{5}) + 9 x^{5} (1 - 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.2.2

Apply the distributive property.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} (1 - 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.2.3

Apply the distributive property.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3

Simplify and combine like terms.

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Step 4.9.3.1

Simplify each term.

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Step 4.9.3.1.1

Multiply $1$ by $1$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.2

Multiply $- 9 x^{5}$ by $1$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.3

Multiply $9$ by $1$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} + 9 x^{5} (- 9 x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5} x^{5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.5

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 4.9.3.1.5.1

Move $x^{5}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 (x^{5} x^{5})))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.5.2

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

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5 + 5}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.5.3

Add $5$ and $5$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.1.6

Multiply $9$ by $- 9$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 9 x^{5} + 9 x^{5} - 81 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.2

Add $- 9 x^{5}$ and $9 x^{5}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 + 0 - 81 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.3.3

Add $1$ and $0$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - (1 - 81 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.4

Apply the distributive property.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 \cdot 1 - (- 81 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.5

Multiply $- 1$ by $1$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 - (- 81 x^{10}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.9.6

Multiply $- 81$ by $- 1$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5})$

Step 4.10

To write $45 x^{4} \arcsin (9 x^{5})$ as a fraction with a common denominator, multiply by $\frac{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + 45 x^{4} \arcsin (9 x^{5}) \cdot \frac{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.11

Write each expression with a common denominator of $(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.11.1

Combine $45 x^{4} \arcsin (9 x^{5})$ and $\frac{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + \frac{45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})}{(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}}$

Step 4.11.2

Reorder the factors of $(1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}$ .

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})} + \frac{45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.12

Combine the numerators over the common denominator.

$\frac{405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + 45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13

Simplify the numerator.

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Step 4.13.1

Factor $45 x^{4}$ out of $405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + 45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})$ .

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Step 4.13.1.1

Factor $45 x^{4}$ out of $405 x^{9} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})$ .

$\frac{45 x^{4} (9 x^{5} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})) + 45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.1.2

Factor $45 x^{4}$ out of $45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}})$ .

$\frac{45 x^{4} (9 x^{5} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})) + 45 x^{4} (\arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.1.3

Factor $45 x^{4}$ out of $45 x^{4} (9 x^{5} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10})) + 45 x^{4} (\arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))$ .

$\frac{45 x^{4} (9 x^{5} (\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 9 x^{5}) (1 - 9 x^{5})}$ as ${((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{1}{2}}$ .

$\frac{45 x^{4} (9 x^{5} ({((1 + 9 x^{5}) (1 - 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3

Simplify each term.

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Step 4.13.3.1

Expand $(1 + 9 x^{5}) (1 - 9 x^{5})$ using the FOIL Method.

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Step 4.13.3.1.1

Apply the distributive property.

$\frac{45 x^{4} (9 x^{5} ({(1 (1 - 9 x^{5}) + 9 x^{5} (1 - 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.1.2

Apply the distributive property.

$\frac{45 x^{4} (9 x^{5} ({(1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} (1 - 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.1.3

Apply the distributive property.

$\frac{45 x^{4} (9 x^{5} ({(1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2

Simplify and combine like terms.

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Step 4.13.3.2.1

Simplify each term.

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Step 4.13.3.2.1.1

Multiply $1$ by $1$ .

$\frac{45 x^{4} (9 x^{5} ({(1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.2

Multiply $- 9 x^{5}$ by $1$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.3

Multiply $9$ by $1$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} + 9 x^{5} (- 9 x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5} x^{5})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.5

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 4.13.3.2.1.5.1

Move $x^{5}$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 (x^{5} x^{5}))}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.5.2

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

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5 + 5})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.5.3

Add $5$ and $5$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{10})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.1.6

Multiply $9$ by $- 9$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 9 x^{5} + 9 x^{5} - 81 x^{10})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.2

Add $- 9 x^{5}$ and $9 x^{5}$ .

$\frac{45 x^{4} (9 x^{5} ({(1 + 0 - 81 x^{10})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.2.3

Add $1$ and $0$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 81 x^{10})}^{\frac{1}{2}} {(1 - 81 x^{10})}^{\frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.3

Multiply ${(1 - 81 x^{10})}^{\frac{1}{2}}$ by ${(1 - 81 x^{10})}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.13.3.3.1

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

$\frac{45 x^{4} (9 x^{5} ({(1 - 81 x^{10})}^{\frac{1}{2} + \frac{1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.3.2

Combine the numerators over the common denominator.

$\frac{45 x^{4} (9 x^{5} ({(1 - 81 x^{10})}^{\frac{1 + 1}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.3.3

Add $1$ and $1$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 81 x^{10})}^{\frac{2}{2}} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.3.4

Divide $2$ by $2$ .

$\frac{45 x^{4} (9 x^{5} ({(1 - 81 x^{10})}^{1} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.3.4

Simplify ${(1 - 81 x^{10})}^{1}$ .

$\frac{45 x^{4} (9 x^{5} (1 - 81 x^{10} - 1 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.4

Combine the opposite terms in $1 - 81 x^{10} - 1 + 81 x^{10}$ .

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Step 4.13.4.1

Subtract $1$ from $1$ .

$\frac{45 x^{4} (9 x^{5} (- 81 x^{10} + 0 + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.4.2

Add $- 81 x^{10}$ and $0$ .

$\frac{45 x^{4} (9 x^{5} (- 81 x^{10} + 81 x^{10}) + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.4.3

Add $- 81 x^{10}$ and $81 x^{10}$ .

$\frac{45 x^{4} (9 x^{5} \cdot 0 + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.5

Multiply $9 x^{5} \cdot 0$ .

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Step 4.13.5.1

Multiply $0$ by $9$ .

$\frac{45 x^{4} (0 x^{5} + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.5.2

Multiply $0$ by $x^{5}$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - 9 x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.6

Expand $(1 + 9 x^{5}) (1 - 9 x^{5})$ using the FOIL Method.

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Step 4.13.6.1

Apply the distributive property.

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 (1 - 9 x^{5}) + 9 x^{5} (1 - 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.6.2

Apply the distributive property.

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} (1 - 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.6.3

Apply the distributive property.

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 \cdot 1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7

Simplify and combine like terms.

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Step 4.13.7.1

Simplify each term.

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Step 4.13.7.1.1

Multiply $1$ by $1$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 + 1 (- 9 x^{5}) + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.2

Multiply $- 9 x^{5}$ by $1$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} \cdot 1 + 9 x^{5} (- 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.3

Multiply $9$ by $1$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} + 9 x^{5} (- 9 x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.4

Rewrite using the commutative property of multiplication.

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5} x^{5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.5

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 4.13.7.1.5.1

Move $x^{5}$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 (x^{5} x^{5})) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.5.2

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

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{5 + 5}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.5.3

Add $5$ and $5$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} + 9 \cdot - 9 x^{10}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.1.6

Multiply $9$ by $- 9$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 9 x^{5} + 9 x^{5} - 81 x^{10}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.2

Add $- 9 x^{5}$ and $9 x^{5}$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 + 0 - 81 x^{10}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.7.3

Add $1$ and $0$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ((1 - 81 x^{10}) {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.8

Multiply $1 - 81 x^{10}$ by ${(1 - 81 x^{10})}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.13.8.1

Multiply $1 - 81 x^{10}$ by ${(1 - 81 x^{10})}^{\frac{1}{2}}$ .

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Step 4.13.8.1.1

Raise $1 - 81 x^{10}$ to the power of $1$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) ({(1 - 81 x^{10})}^{1} {(1 - 81 x^{10})}^{\frac{1}{2}}))}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.8.1.2

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

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{1 + \frac{1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.8.2

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

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{2}{2} + \frac{1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.8.3

Combine the numerators over the common denominator.

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{2 + 1}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.8.4

Add $2$ and $1$ .

$\frac{45 x^{4} (0 + \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{3}{2}})}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.13.9

Add $0$ and $\arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{3}{2}}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{3}{2}}}{{(1 - 81 x^{10})}^{\frac{1}{2}} (1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.14

Move ${(1 - 81 x^{10})}^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{3}{2}} {(1 - 81 x^{10})}^{- \frac{1}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15

Simplify the numerator.

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Step 4.15.1

Multiply ${(1 - 81 x^{10})}^{\frac{3}{2}}$ by ${(1 - 81 x^{10})}^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.15.1.1

Move ${(1 - 81 x^{10})}^{- \frac{1}{2}}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) ({(1 - 81 x^{10})}^{- \frac{1}{2}} {(1 - 81 x^{10})}^{\frac{3}{2}})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15.1.2

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

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{- \frac{1}{2} + \frac{3}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15.1.3

Combine the numerators over the common denominator.

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{- 1 + 3}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15.1.4

Add $- 1$ and $3$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{\frac{2}{2}}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15.1.5

Divide $2$ by $2$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{1}}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.15.2

Simplify $45 x^{4} \arcsin (9 x^{5}) {(1 - 81 x^{10})}^{1}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) (1 - 81 x^{10})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.16

Simplify the numerator.

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Step 4.16.1

Rewrite $1$ as $1^{2}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) (1^{2} - 81 x^{10})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.16.2

Rewrite $81 x^{10}$ as ${(9 x^{5})}^{2}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) (1^{2} - {(9 x^{5})}^{2})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.16.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 9 x^{5}$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) ((1 + 9 x^{5}) (1 - (9 x^{5})))}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.16.4

Multiply $9$ by $- 1$ .

$\frac{45 x^{4} \arcsin (9 x^{5}) (1 + 9 x^{5}) (1 - 9 x^{5})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.17

Cancel the common factor of $1 + 9 x^{5}$ .

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Step 4.17.1

Cancel the common factor.

$\frac{45 x^{4} \arcsin (9 x^{5}) (1 + 9 x^{5}) (1 - 9 x^{5})}{(1 + 9 x^{5}) (1 - 9 x^{5})}$

Step 4.17.2

Rewrite the expression.

$\frac{45 x^{4} \arcsin (9 x^{5}) (1 - 9 x^{5})}{1 - 9 x^{5}}$

Step 4.18

Cancel the common factor of $1 - 9 x^{5}$ .

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Step 4.18.1

Cancel the common factor.

$\frac{45 x^{4} \arcsin (9 x^{5}) (1 - 9 x^{5})}{1 - 9 x^{5}}$

Step 4.18.2

Divide $45 x^{4} \arcsin (9 x^{5})$ by $1$ .

$45 x^{4} \arcsin (9 x^{5})$