Calculus Examples

$f (x) = 3 x \sqrt{2 - x^{2}} - \frac{3 (x^{2})}{{(2 - x^{2})}^{\frac{1}{2}}}$

Step 1

By the Sum Rule, the derivative of $3 x \sqrt{2 - x^{2}} - \frac{3 (x^{2})}{{(2 - x^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [3 x \sqrt{2 - x^{2}}] + \frac{d}{d x} [- \frac{3 (x^{2})}{{(2 - x^{2})}^{\frac{1}{2}}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [3 x \sqrt{2 - x^{2}}]$ .

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Step 2.1

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

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

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x {(2 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x {(2 - x^{2})}^{\frac{1}{2}}]$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Combine the numerators over the common denominator.

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

Step 2.13

Simplify the numerator.

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Step 2.13.1

Multiply $- 1$ by $2$ .

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

Step 2.13.2

Subtract $2$ from $1$ .

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

Step 2.14

Move the negative in front of the fraction.

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

Step 2.15

Multiply $2$ by $- 1$ .

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

Step 2.16

Subtract $2 x$ from $0$ .

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

Factor $2$ out of $- 2 x$ .

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

Step 2.22

Cancel the common factors.

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Step 2.22.1

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

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

Step 2.22.2

Cancel the common factor.

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

Step 2.22.3

Rewrite the expression.

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

Step 2.23

Move the negative in front of the fraction.

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

Step 2.24

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

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

Step 2.25

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

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

Step 2.26

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

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

Step 2.27

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

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

Step 2.28

Add $1$ and $1$ .

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

Step 2.29

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

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

Step 2.30

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

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

Step 2.31

Combine the numerators over the common denominator.

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

Step 2.32

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

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Step 2.32.1

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

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

Step 2.32.2

Combine the numerators over the common denominator.

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

Step 2.32.3

Add $1$ and $1$ .

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

Step 2.32.4

Divide $2$ by $2$ .

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

Step 2.33

Simplify ${(2 - x^{2})}^{1}$ .

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

Step 2.34

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

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

Step 2.35

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

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{3 (x^{2})}{{(2 - x^{2})}^{\frac{1}{2}}}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3 x^{2}}{{(2 - x^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{x^{2}}{{(2 - x^{2})}^{\frac{1}{2}}}]$ .

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

Step 3.2

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 $g (x) = {(2 - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

$\frac{3 (- 2 x^{2} + 2)}{{(2 - x^{2})}^{\frac{1}{2}}} - 3 \frac{2 \cdot {(2 - x^{2})}^{\frac{1}{2}} x - x^{2} (\frac{1}{2} {(2 - x^{2})}^{\frac{1}{2} - 1} (0 - (2 x)))}{{({(2 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.10

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

$\frac{3 (- 2 x^{2} + 2)}{{(2 - x^{2})}^{\frac{1}{2}}} - 3 \frac{2 {(2 - x^{2})}^{\frac{1}{2}} x - x^{2} (\frac{1}{2} {(2 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - (2 x)))}{{({(2 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.11

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

$\frac{3 (- 2 x^{2} + 2)}{{(2 - x^{2})}^{\frac{1}{2}}} - 3 \frac{2 {(2 - x^{2})}^{\frac{1}{2}} x - x^{2} (\frac{1}{2} {(2 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - (2 x)))}{{({(2 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.12

Combine the numerators over the common denominator.

$\frac{3 (- 2 x^{2} + 2)}{{(2 - x^{2})}^{\frac{1}{2}}} - 3 \frac{2 {(2 - x^{2})}^{\frac{1}{2}} x - x^{2} (\frac{1}{2} {(2 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 - (2 x)))}{{({(2 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $- 1$ by $2$ .

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

Step 3.13.2

Subtract $2$ from $1$ .

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

Step 3.14

Move the negative in front of the fraction.

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

Step 3.15

Multiply $2$ by $- 1$ .

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

Step 3.16

Subtract $2 x$ from $0$ .

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

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

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

Step 3.21

Factor $2$ out of $- 2 x$ .

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

Step 3.22

Cancel the common factors.

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Step 3.22.1

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

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

Step 3.22.2

Cancel the common factor.

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

Step 3.22.3

Rewrite the expression.

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

Step 3.23

Move the negative in front of the fraction.

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

Step 3.24

Multiply $- 1$ by $- 1$ .

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

Step 3.25

Multiply $x^{2}$ by $1$ .

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

Step 3.26

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

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

Step 3.27

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.27.1

Multiply $x^{2}$ by $x$ .

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Step 3.27.1.1

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

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

Step 3.27.1.2

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

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

Step 3.27.2

Add $2$ and $1$ .

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

Step 3.28

Reorder $2$ and $- x^{2}$ .

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

Step 3.29

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

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

Step 3.30

Combine the numerators over the common denominator.

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

Step 3.31

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

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Step 3.31.1

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

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

Step 3.31.2

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

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

Step 3.31.3

Combine the numerators over the common denominator.

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

Step 3.31.4

Add $1$ and $1$ .

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

Step 3.31.5

Divide $2$ by $2$ .

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

Step 3.32

Simplify $2 x {(- x^{2} + 2)}^{1}$ .

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

Step 3.33

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

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Step 3.33.1

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

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

Step 3.33.2

Cancel the common factor of $2$ .

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Step 3.33.2.1

Cancel the common factor.

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

Step 3.33.2.2

Rewrite the expression.

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

Step 3.34

Simplify.

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

Step 3.35

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

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

Step 3.36

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

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

Step 3.37

Reorder terms.

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

Step 3.38

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

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Step 3.38.1

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

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Step 3.38.1.1

Raise $- x^{2} + 2$ to the power of $1$ .

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

Step 3.38.1.2

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

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

Step 3.38.2

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

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

Step 3.38.3

Combine the numerators over the common denominator.

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

Step 3.38.4

Add $1$ and $2$ .

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

Step 3.39

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

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

Step 3.40

Move the negative in front of the fraction.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $- 2$ by $3$ .

$\frac{- 6 x^{2} + 3 \cdot 2}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (2 x (- x^{2})) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.2

Multiply $3$ by $2$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (2 x (- x^{2})) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.3

Multiply $- 1$ by $2$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (- 2 x \cdot x^{2}) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.4.4.1

Move $x^{2}$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (- 2 (x^{2} x)) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.4.2

Multiply $x^{2}$ by $x$ .

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Step 4.4.4.2.1

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

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (- 2 (x^{2} x^{1})) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.4.2.2

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

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (- 2 x^{2 + 1}) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.4.3

Add $2$ and $1$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{3 (- 2 x^{3}) + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.5

Multiply $- 2$ by $3$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{- 6 x^{3} + 3 (2 x \cdot 2) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.6

Multiply $2$ by $2$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{- 6 x^{3} + 3 (4 x) + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.7

Multiply $4$ by $3$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{- 6 x^{3} + 12 x + 3 x^{3}}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.8

Add $- 6 x^{3}$ and $3 x^{3}$ .

$\frac{- 6 x^{2} + 6}{{(2 - x^{2})}^{\frac{1}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.9

Reorder terms.

$\frac{- 6 x^{2} + 6}{{(- x^{2} + 2)}^{\frac{1}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.10

To write $\frac{- 6 x^{2} + 6}{{(- x^{2} + 2)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{2}{2}}}$ .

$\frac{- 6 x^{2} + 6}{{(- x^{2} + 2)}^{\frac{1}{2}}} \cdot \frac{{(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{2}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.11

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

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Step 4.4.11.1

Multiply $\frac{- 6 x^{2} + 6}{{(- x^{2} + 2)}^{\frac{1}{2}}}$ by $\frac{{(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{2}{2}}}$ .

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{1}{2}} {(- x^{2} + 2)}^{\frac{2}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.11.2

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

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Step 4.4.11.2.1

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

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{1}{2} + \frac{2}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.11.2.2

Combine the numerators over the common denominator.

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{1 + 2}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.11.2.3

Add $1$ and $2$ .

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}}}{{(- x^{2} + 2)}^{\frac{3}{2}}} - \frac{- 3 x^{3} + 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.12

Combine the numerators over the common denominator.

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}} - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.13

Cancel the common factor of $2$ .

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Step 4.4.13.1

Cancel the common factor.

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{\frac{2}{2}} - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.13.2

Rewrite the expression.

$\frac{(- 6 x^{2} + 6) {(- x^{2} + 2)}^{1} - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.4.14

Simplify.

$\frac{(- 6 x^{2} + 6) (- x^{2} + 2) - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Expand $(- 6 x^{2} + 6) (- x^{2} + 2)$ using the FOIL Method.

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Step 4.5.1.1

Apply the distributive property.

$\frac{- 6 x^{2} (- x^{2} + 2) + 6 (- x^{2} + 2) - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.1.2

Apply the distributive property.

$\frac{- 6 x^{2} (- x^{2}) - 6 x^{2} \cdot 2 + 6 (- x^{2} + 2) - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.1.3

Apply the distributive property.

$\frac{- 6 x^{2} (- x^{2}) - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2

Simplify and combine like terms.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 6 \cdot - 1 x^{2} x^{2} - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.5.2.1.2.1

Move $x^{2}$ .

$\frac{- 6 \cdot - 1 (x^{2} x^{2}) - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.2.2

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

$\frac{- 6 \cdot - 1 x^{2 + 2} - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.2.3

Add $2$ and $2$ .

$\frac{- 6 \cdot - 1 x^{4} - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.3

Multiply $- 6$ by $- 1$ .

$\frac{6 x^{4} - 6 x^{2} \cdot 2 + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.4

Multiply $2$ by $- 6$ .

$\frac{6 x^{4} - 12 x^{2} + 6 (- x^{2}) + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.5

Multiply $- 1$ by $6$ .

$\frac{6 x^{4} - 12 x^{2} - 6 x^{2} + 6 \cdot 2 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.1.6

Multiply $6$ by $2$ .

$\frac{6 x^{4} - 12 x^{2} - 6 x^{2} + 12 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.2.2

Subtract $6 x^{2}$ from $- 12 x^{2}$ .

$\frac{6 x^{4} - 18 x^{2} + 12 - (- 3 x^{3} + 12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.3

Apply the distributive property.

$\frac{6 x^{4} - 18 x^{2} + 12 - (- 3 x^{3}) - (12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.4

Multiply $- 3$ by $- 1$ .

$\frac{6 x^{4} - 18 x^{2} + 12 + 3 x^{3} - (12 x)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.5

Multiply $12$ by $- 1$ .

$\frac{6 x^{4} - 18 x^{2} + 12 + 3 x^{3} - 12 x}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.6

Reorder terms.

$\frac{6 x^{4} + 3 x^{3} - 18 x^{2} - 12 x + 12}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7

Factor $3$ out of $6 x^{4} + 3 x^{3} - 18 x^{2} - 12 x + 12$ .

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Step 4.5.7.1

Factor $3$ out of $6 x^{4}$ .

$\frac{3 (2 x^{4}) + 3 x^{3} - 18 x^{2} - 12 x + 12}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.2

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

$\frac{3 (2 x^{4}) + 3 (x^{3}) - 18 x^{2} - 12 x + 12}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.3

Factor $3$ out of $- 18 x^{2}$ .

$\frac{3 (2 x^{4}) + 3 (x^{3}) + 3 (- 6 x^{2}) - 12 x + 12}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.4

Factor $3$ out of $- 12 x$ .

$\frac{3 (2 x^{4}) + 3 (x^{3}) + 3 (- 6 x^{2}) + 3 (- 4 x) + 12}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.5

Factor $3$ out of $12$ .

$\frac{3 (2 x^{4}) + 3 x^{3} + 3 (- 6 x^{2}) + 3 (- 4 x) + 3 \cdot 4}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.6

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

$\frac{3 (2 x^{4} + x^{3}) + 3 (- 6 x^{2}) + 3 (- 4 x) + 3 \cdot 4}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.7

Factor $3$ out of $3 (2 x^{4} + x^{3}) + 3 (- 6 x^{2})$ .

$\frac{3 (2 x^{4} + x^{3} - 6 x^{2}) + 3 (- 4 x) + 3 \cdot 4}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.8

Factor $3$ out of $3 (2 x^{4} + x^{3} - 6 x^{2}) + 3 (- 4 x)$ .

$\frac{3 (2 x^{4} + x^{3} - 6 x^{2} - 4 x) + 3 \cdot 4}{{(- x^{2} + 2)}^{\frac{3}{2}}}$

Step 4.5.7.9

Factor $3$ out of $3 (2 x^{4} + x^{3} - 6 x^{2} - 4 x) + 3 \cdot 4$ .

$\frac{3 (2 x^{4} + x^{3} - 6 x^{2} - 4 x + 4)}{{(- x^{2} + 2)}^{\frac{3}{2}}}$