Calculus Examples

$\frac{\sqrt{5 - x^{3}}}{\sqrt{4 + x^{3}}}$

Step 1

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 $f (x) = \sqrt{5 - x^{3}}$ and $g (x) = \sqrt{4 + x^{3}}$ .

$\frac{\sqrt{4 + x^{3}} \frac{d}{d x} [\sqrt{5 - x^{3}}] - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 2

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

$\frac{\sqrt{4 + x^{3}} \frac{d}{d x} [{(5 - x^{3})}^{\frac{1}{2}}] - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 3

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

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Step 3.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 3.3

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

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2} {(5 - x^{3})}^{\frac{1}{2} - 1} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2} {(5 - x^{3})}^{\frac{1 - 2}{2}} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 7.2

Subtract $2$ from $1$ .

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2} {(5 - x^{3})}^{\frac{- 1}{2}} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8

Differentiate.

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Step 8.1

Move the negative in front of the fraction.

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2} {(5 - x^{3})}^{- \frac{1}{2}} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.2

Combine fractions.

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Step 8.2.1

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

$\frac{\sqrt{4 + x^{3}} (\frac{{(5 - x^{3})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.2.2

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

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [5 - x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.3

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

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} (\frac{d}{d x} [5] + \frac{d}{d x} [- x^{3}])) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.4

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

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

Step 8.5

Add $0$ and $\frac{d}{d x} [- x^{3}]$ .

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{3}]) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{3}])) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.7

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

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} (- (3 x^{2}))) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.8

Combine fractions.

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Step 8.8.1

Multiply $3$ by $- 1$ .

$\frac{\sqrt{4 + x^{3}} (\frac{1}{2 {(5 - x^{3})}^{\frac{1}{2}}} (- 3 x^{2})) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.8.2

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

$\frac{\sqrt{4 + x^{3}} (\frac{- 3}{2 {(5 - x^{3})}^{\frac{1}{2}}} x^{2}) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.8.3

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

$\frac{\sqrt{4 + x^{3}} \frac{- 3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}} - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.8.4

Simplify the expression.

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Step 8.8.4.1

Move the negative in front of the fraction.

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} \frac{d}{d x} [\sqrt{4 + x^{3}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 8.8.4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + x^{3}}$ as ${(4 + x^{3})}^{\frac{1}{2}}$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} \frac{d}{d x} [{(4 + x^{3})}^{\frac{1}{2}}]}{{\sqrt{4 + x^{3}}}^{2}}$

Step 9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 + x^{3}$ .

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Step 9.1

To apply the Chain Rule, set $u_{2}$ as $4 + x^{3}$ .

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

Step 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 $n = \frac{1}{2}$ .

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

Step 9.3

Replace all occurrences of $u_{2}$ with $4 + x^{3}$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2} {(4 + x^{3})}^{\frac{1}{2} - 1} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2} {(4 + x^{3})}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 13.2

Subtract $2$ from $1$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2} {(4 + x^{3})}^{\frac{- 1}{2}} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 14

Combine fractions.

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Step 14.1

Move the negative in front of the fraction.

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2} {(4 + x^{3})}^{- \frac{1}{2}} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 14.2

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

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{{(4 + x^{3})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 14.3

Move ${(4 + x^{3})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2 {(4 + x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [4 + x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 15

By the Sum Rule, the derivative of $4 + x^{3}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [x^{3}]$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2 {(4 + x^{3})}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [x^{3}]))}{{\sqrt{4 + x^{3}}}^{2}}$

Step 16

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

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

Step 17

Add $0$ and $\frac{d}{d x} [x^{3}]$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2 {(4 + x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}])}{{\sqrt{4 + x^{3}}}^{2}}$

Step 18

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

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{1}{2 {(4 + x^{3})}^{\frac{1}{2}}} (3 x^{2}))}{{\sqrt{4 + x^{3}}}^{2}}$

Step 19

Combine fractions.

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Step 19.1

Combine $3$ and $\frac{1}{2 {(4 + x^{3})}^{\frac{1}{2}}}$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} (\frac{3}{2 {(4 + x^{3})}^{\frac{1}{2}}} x^{2})}{{\sqrt{4 + x^{3}}}^{2}}$

Step 19.2

Combine $\frac{3}{2 {(4 + x^{3})}^{\frac{1}{2}}}$ and $x^{2}$ .

$\frac{\sqrt{4 + x^{3}} (- \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}) - \sqrt{5 - x^{3}} \frac{3 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20

Simplify.

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Step 20.1

Simplify the numerator.

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Step 20.1.1

Rewrite using the commutative property of multiplication.

$\frac{- \sqrt{4 + x^{3}} \frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}} - \sqrt{5 - x^{3}} \frac{3 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.2

Combine $\frac{3 x^{2}}{2 {(5 - x^{3})}^{\frac{1}{2}}}$ and $\sqrt{4 + x^{3}}$ .

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}}}{2 {(5 - x^{3})}^{\frac{1}{2}}} - \sqrt{5 - x^{3}} \frac{3 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.3

Combine $\frac{3 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}}}$ and $\sqrt{5 - x^{3}}$ .

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}}}{2 {(5 - x^{3})}^{\frac{1}{2}}} - \frac{3 x^{2} \sqrt{5 - x^{3}}}{2 {(4 + x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.4

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

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}}}{2 {(5 - x^{3})}^{\frac{1}{2}}} \cdot \frac{{(4 + x^{3})}^{\frac{1}{2}}}{{(4 + x^{3})}^{\frac{1}{2}}} - \frac{3 x^{2} \sqrt{5 - x^{3}}}{2 {(4 + x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.5

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

Step 20.1.6

Write each expression with a common denominator of $2 {(5 - x^{3})}^{\frac{1}{2}} {(4 + x^{3})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 20.1.6.1

Multiply $\frac{3 x^{2} \sqrt{4 + x^{3}}}{2 {(5 - x^{3})}^{\frac{1}{2}}}$ by $\frac{{(4 + x^{3})}^{\frac{1}{2}}}{{(4 + x^{3})}^{\frac{1}{2}}}$ .

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}}{2 {(5 - x^{3})}^{\frac{1}{2}} {(4 + x^{3})}^{\frac{1}{2}}} - \frac{3 x^{2} \sqrt{5 - x^{3}}}{2 {(4 + x^{3})}^{\frac{1}{2}}} \cdot \frac{{(5 - x^{3})}^{\frac{1}{2}}}{{(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.6.2

Multiply $\frac{3 x^{2} \sqrt{5 - x^{3}}}{2 {(4 + x^{3})}^{\frac{1}{2}}}$ by $\frac{{(5 - x^{3})}^{\frac{1}{2}}}{{(5 - x^{3})}^{\frac{1}{2}}}$ .

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}}{2 {(5 - x^{3})}^{\frac{1}{2}} {(4 + x^{3})}^{\frac{1}{2}}} - \frac{3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.6.3

Reorder the factors of $2 {(5 - x^{3})}^{\frac{1}{2}} {(4 + x^{3})}^{\frac{1}{2}}$ .

$\frac{- \frac{3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}} - \frac{3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.7

Combine the numerators over the common denominator.

$\frac{\frac{- 3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}} - 3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8

Rewrite $\frac{- 3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}} - 3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$ in a factored form.

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Step 20.1.8.1

Factor $3 x^{2}$ out of $- 3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}} - 3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}$ .

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Step 20.1.8.1.1

Factor $3 x^{2}$ out of $- 3 x^{2} \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}$ .

$\frac{\frac{3 x^{2} (- \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}) - 3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.1.2

Factor $3 x^{2}$ out of $- 3 x^{2} \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}}$ .

$\frac{\frac{3 x^{2} (- \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}) + 3 x^{2} (- \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.1.3

Factor $3 x^{2}$ out of $3 x^{2} (- \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}}) + 3 x^{2} (- \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})$ .

$\frac{\frac{3 x^{2} (- \sqrt{4 + x^{3}} {(4 + x^{3})}^{\frac{1}{2}} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2

Combine exponents.

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Step 20.1.8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + x^{3}}$ as ${(4 + x^{3})}^{\frac{1}{2}}$ .

$\frac{\frac{3 x^{2} (- ({(4 + x^{3})}^{\frac{1}{2}} {(4 + x^{3})}^{\frac{1}{2}}) - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.2

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

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{\frac{1}{2} + \frac{1}{2}} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{\frac{1 + 1}{2}} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.4

Add $1$ and $1$ .

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{\frac{2}{2}} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.5

Cancel the common factor of $2$ .

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Step 20.1.8.2.5.1

Cancel the common factor.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{\frac{2}{2}} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.5.2

Rewrite the expression.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - \sqrt{5 - x^{3}} {(5 - x^{3})}^{\frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.6

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

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - ({(5 - x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}))}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.7

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

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - {(5 - x^{3})}^{\frac{1}{2} + \frac{1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.8

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - {(5 - x^{3})}^{\frac{1 + 1}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.9

Add $1$ and $1$ .

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - {(5 - x^{3})}^{\frac{2}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.10

Cancel the common factor of $2$ .

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Step 20.1.8.2.10.1

Cancel the common factor.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - {(5 - x^{3})}^{\frac{2}{2}})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.8.2.10.2

Rewrite the expression.

$\frac{\frac{3 x^{2} (- {(4 + x^{3})}^{1} - {(5 - x^{3})}^{1})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9

Simplify the numerator.

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Step 20.1.9.1

Simplify.

$\frac{\frac{3 x^{2} (- (4 + x^{3}) - {(5 - x^{3})}^{1})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.2

Apply the distributive property.

$\frac{\frac{3 x^{2} (- 1 \cdot 4 - x^{3} - {(5 - x^{3})}^{1})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.3

Multiply $- 1$ by $4$ .

$\frac{\frac{3 x^{2} (- 4 - x^{3} - {(5 - x^{3})}^{1})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.4

Simplify.

$\frac{\frac{3 x^{2} (- 4 - x^{3} - (5 - x^{3}))}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.5

Apply the distributive property.

$\frac{\frac{3 x^{2} (- 4 - x^{3} - 1 \cdot 5 - - x^{3})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.6

Multiply $- 1$ by $5$ .

$\frac{\frac{3 x^{2} (- 4 - x^{3} - 5 - - x^{3})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.7

Multiply $- - x^{3}$ .

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Step 20.1.9.7.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{3 x^{2} (- 4 - x^{3} - 5 + 1 x^{3})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.7.2

Multiply $x^{3}$ by $1$ .

$\frac{\frac{3 x^{2} (- 4 - x^{3} - 5 + x^{3})}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.9.8

Subtract $5$ from $- 4$ .

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

Step 20.1.9.9

Add $- x^{3}$ and $x^{3}$ .

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

Step 20.1.9.10

Subtract $9$ from $0$ .

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

Step 20.1.9.11

Multiply $- 9$ by $3$ .

$\frac{\frac{- 27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.1.10

Move the negative in front of the fraction.

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{\sqrt{4 + x^{3}}}^{2}}$

Step 20.2

Combine terms.

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Step 20.2.1

Rewrite ${\sqrt{4 + x^{3}}}^{2}$ as $4 + x^{3}$ .

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Step 20.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + x^{3}}$ as ${(4 + x^{3})}^{\frac{1}{2}}$ .

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{({(4 + x^{3})}^{\frac{1}{2}})}^{2}}$

Step 20.2.1.2

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

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{(4 + x^{3})}^{\frac{1}{2} \cdot 2}}$

Step 20.2.1.3

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

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{(4 + x^{3})}^{\frac{2}{2}}}$

Step 20.2.1.4

Cancel the common factor of $2$ .

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Step 20.2.1.4.1

Cancel the common factor.

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{(4 + x^{3})}^{\frac{2}{2}}}$

Step 20.2.1.4.2

Rewrite the expression.

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{{(4 + x^{3})}^{1}}$

Step 20.2.1.5

Simplify.

$\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{4 + x^{3}}$

Step 20.2.2

Rewrite $\frac{- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}}{4 + x^{3}}$ as a product.

$- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}} \cdot \frac{1}{4 + x^{3}}$

Step 20.2.3

Multiply $\frac{1}{4 + x^{3}}$ by $\frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$ .

$- \frac{27 x^{2}}{(4 + x^{3}) (2 {(4 + x^{3})}^{\frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}})}$

Step 20.2.4

Raise $4 + x^{3}$ to the power of $1$ .

$- \frac{27 x^{2}}{2 ({(4 + x^{3})}^{1} {(4 + x^{3})}^{\frac{1}{2}}) {(5 - x^{3})}^{\frac{1}{2}}}$

Step 20.2.5

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

$- \frac{27 x^{2}}{2 {(4 + x^{3})}^{1 + \frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$

Step 20.2.6

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

$- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{2}{2} + \frac{1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$

Step 20.2.7

Combine the numerators over the common denominator.

$- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{2 + 1}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$

Step 20.2.8

Add $2$ and $1$ .

$- \frac{27 x^{2}}{2 {(4 + x^{3})}^{\frac{3}{2}} {(5 - x^{3})}^{\frac{1}{2}}}$