Calculus Examples

$y = \frac{x^{2}}{x^{2} + 6}$

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = x^{2}$ and $g (x) = x^{2} + 6$ .

$f' (x) = \frac{(x^{2} + 6) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x^{2} + 6)}{{(x^{2} + 6)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$f' (x) = \frac{(x^{2} + 6) (2 x) - x^{2} \frac{d}{d x} (x^{2} + 6)}{{(x^{2} + 6)}^{2}}$

Step 1.2.2

Move $2$ to the left of $x^{2} + 6$ .

$f' (x) = \frac{2 \cdot ((x^{2} + 6) x) - x^{2} \frac{d}{d x} (x^{2} + 6)}{{(x^{2} + 6)}^{2}}$

Step 1.2.3

By the Sum Rule, the derivative of $x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]$ .

$f' (x) = \frac{2 (x^{2} + 6) x - x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{2}}$

Step 1.2.4

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

$f' (x) = \frac{2 (x^{2} + 6) x - x^{2} (2 x + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{2}}$

Step 1.2.5

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} + 6) x - x^{2} (2 x + 0)}{{(x^{2} + 6)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{2 (x^{2} + 6) x - x^{2} (2 x)}{{(x^{2} + 6)}^{2}}$

Step 1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{2 (x^{2} + 6) x - 2 x^{2} x}{{(x^{2} + 6)}^{2}}$

Step 1.3

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

$f' (x) = \frac{2 (x^{2} + 6) x - 2 (x \cdot x^{2})}{{(x^{2} + 6)}^{2}}$

Step 1.4

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

$f' (x) = \frac{2 (x^{2} + 6) x - 2 x^{1 + 2}}{{(x^{2} + 6)}^{2}}$

Step 1.5

Add $1$ and $2$ .

$f' (x) = \frac{2 (x^{2} + 6) x - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 \cdot 6) x - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6.2

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x + 2 \cdot (6 x) - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6.3

Simplify the numerator.

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Step 1.6.3.1

Simplify each term.

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Step 1.6.3.1.1

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

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Step 1.6.3.1.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (6 x) - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6.3.1.1.2

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

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Step 1.6.3.1.1.2.1

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

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (6 x) - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6.3.1.1.2.2

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

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

Step 1.6.3.1.1.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} + 2 \cdot (6 x) - 2 x^{3}}{{(x^{2} + 6)}^{2}}$

Step 1.6.3.1.2

Multiply $2$ by $6$ .

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

Step 1.6.3.2

Combine the opposite terms in $2 x^{3} + 12 x - 2 x^{3}$ .

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Step 1.6.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{12 x + 0}{{(x^{2} + 6)}^{2}}$

Step 1.6.3.2.2

Add $12 x$ and $0$ .

$f' (x) = \frac{12 x}{{(x^{2} + 6)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $\frac{12 x}{{(x^{2} + 6)}^{2}}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{x}{{(x^{2} + 6)}^{2}}]$ .

$f'' (x) = 12 \frac{d}{d x} (\frac{x}{{(x^{2} + 6)}^{2}})$

Step 2.2

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 6)}^{2}}{{({(x^{2} + 6)}^{2})}^{2}})$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${({(x^{2} + 6)}^{2})}^{2}$ .

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Step 2.3.1.1

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

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 6)}^{2}}{{(x^{2} + 6)}^{2 \cdot 2}})$

Step 2.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 6)}^{2}}{{(x^{2} + 6)}^{4}})$

Step 2.3.2

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

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} + 6)}^{2}}{{(x^{2} + 6)}^{4}})$

Step 2.3.3

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

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} - x \frac{d}{d x} {(x^{2} + 6)}^{2}}{{(x^{2} + 6)}^{4}})$

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 $f (x) = x^{2}$ and $g (x) = x^{2} + 6$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 6$ .

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}})$

Step 2.4.2

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

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} - x (2 u \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}})$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 6$ .

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} - x (2 (x^{2} + 6) \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}})$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = 12 (\frac{{(x^{2} + 6)}^{2} - 2 x ((x^{2} + 6) \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}})$

Step 2.5.2

Factor $x^{2} + 6$ out of ${(x^{2} + 6)}^{2} - 2 x ((x^{2} + 6) \frac{d}{d x} [x^{2} + 6])$ .

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Step 2.5.2.1

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

$f'' (x) = 12 (\frac{(x^{2} + 6) (x^{2} + 6) - 2 x ((x^{2} + 6) \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}})$

Step 2.5.2.2

Factor $x^{2} + 6$ out of $- 2 x ((x^{2} + 6) \frac{d}{d x} [x^{2} + 6])$ .

$f'' (x) = 12 (\frac{(x^{2} + 6) (x^{2} + 6) + (x^{2} + 6) (- 2 x (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{4}})$

Step 2.5.2.3

Factor $x^{2} + 6$ out of $(x^{2} + 6) (x^{2} + 6) + (x^{2} + 6) (- 2 x (\frac{d}{d x} [x^{2} + 6]))$ .

$f'' (x) = 12 (\frac{(x^{2} + 6) (x^{2} + 6 - 2 x (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{4}})$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$f'' (x) = 12 (\frac{(x^{2} + 6) (x^{2} + 6 - 2 x (\frac{d}{d x} (x^{2} + 6)))}{(x^{2} + 6) {(x^{2} + 6)}^{3}})$

Step 2.6.2

Cancel the common factor.

$f'' (x) = 12 (\frac{(x^{2} + 6) (x^{2} + 6 - 2 x (\frac{d}{d x} (x^{2} + 6)))}{(x^{2} + 6) {(x^{2} + 6)}^{3}})$

Step 2.6.3

Rewrite the expression.

$f'' (x) = 12 (\frac{x^{2} + 6 - 2 x (\frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{3}})$

Step 2.7

By the Sum Rule, the derivative of $x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]$ .

$f'' (x) = 12 (\frac{x^{2} + 6 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{3}})$

Step 2.8

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

$f'' (x) = 12 (\frac{x^{2} + 6 - 2 x (2 x + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{3}})$

Step 2.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f'' (x) = 12 (\frac{x^{2} + 6 - 2 x (2 x + 0)}{{(x^{2} + 6)}^{3}})$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 2$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Add $1$ and $1$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Simplify.

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Step 2.17.1

Apply the distributive property.

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

Step 2.17.2

Simplify each term.

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Step 2.17.2.1

Multiply $- 3$ by $12$ .

$f'' (x) = \frac{- 36 x^{2} + 12 \cdot 6}{{(x^{2} + 6)}^{3}}$

Step 2.17.2.2

Multiply $12$ by $6$ .

$f'' (x) = \frac{- 36 x^{2} + 72}{{(x^{2} + 6)}^{3}}$

Step 2.17.3

Factor $36$ out of $- 36 x^{2} + 72$ .

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Step 2.17.3.1

Factor $36$ out of $- 36 x^{2}$ .

$f'' (x) = \frac{36 (- x^{2}) + 72}{{(x^{2} + 6)}^{3}}$

Step 2.17.3.2

Factor $36$ out of $72$ .

$f'' (x) = \frac{36 (- x^{2}) + 36 (2)}{{(x^{2} + 6)}^{3}}$

Step 2.17.3.3

Factor $36$ out of $36 (- x^{2}) + 36 (2)$ .

$f'' (x) = \frac{36 (- x^{2} + 2)}{{(x^{2} + 6)}^{3}}$

Step 2.17.4

Factor $- 1$ out of $- x^{2}$ .

$f'' (x) = \frac{36 (- (x^{2}) + 2)}{{(x^{2} + 6)}^{3}}$

Step 2.17.5

Rewrite $2$ as $- 1 (- 2)$ .

$f'' (x) = \frac{36 (- (x^{2}) - 1 \cdot - 2)}{{(x^{2} + 6)}^{3}}$

Step 2.17.6

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

$f'' (x) = \frac{36 (- (x^{2} - 2))}{{(x^{2} + 6)}^{3}}$

Step 2.17.7

Rewrite $- (x^{2} - 2)$ as $- 1 (x^{2} - 2)$ .

$f'' (x) = \frac{36 (- 1 (x^{2} - 2))}{{(x^{2} + 6)}^{3}}$

Step 2.17.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{36 (x^{2} - 2)}{{(x^{2} + 6)}^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- \frac{36 (x^{2} - 2)}{{(x^{2} + 6)}^{3}}$ with respect to $x$ is $- 36 \frac{d}{d x} [\frac{x^{2} - 2}{{(x^{2} + 6)}^{3}}]$ .

$f''' (x) = - 36 \frac{d}{d x} \frac{x^{2} - 2}{{(x^{2} + 6)}^{3}}$

Step 3.2

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} \frac{d}{d x} (x^{2} - 2) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{({(x^{2} + 6)}^{3})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x^{2} + 6)}^{3})}^{2}$ .

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Step 3.3.1.1

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} \frac{d}{d x} (x^{2} - 2) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{3 \cdot 2}}$

Step 3.3.1.2

Multiply $3$ by $2$ .

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} \frac{d}{d x} (x^{2} - 2) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

Step 3.3.2

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2)) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

Step 3.3.3

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} (2 x + \frac{d}{d x} (- 2)) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

Step 3.3.4

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} (2 x + 0) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $2 x$ and $0$ .

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{3} (2 x) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

Step 3.3.5.2

Move $2$ to the left of ${(x^{2} + 6)}^{3}$ .

$f''' (x) = - 36 \frac{2 \cdot ({(x^{2} + 6)}^{3} x) - (x^{2} - 2) \frac{d}{d x} {(x^{2} + 6)}^{3}}{{(x^{2} + 6)}^{6}}$

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 $f (x) = x^{3}$ and $g (x) = x^{2} + 6$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 6$ .

$f''' (x) = - 36 \frac{2 {(x^{2} + 6)}^{3} x - (x^{2} - 2) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{6}}$

Step 3.4.2

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

$f''' (x) = - 36 \frac{2 {(x^{2} + 6)}^{3} x - (x^{2} - 2) (3 u^{2} \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{6}}$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} + 6$ .

$f''' (x) = - 36 \frac{2 {(x^{2} + 6)}^{3} x - (x^{2} - 2) (3 {(x^{2} + 6)}^{2} \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{6}}$

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$f''' (x) = - 36 \frac{2 {(x^{2} + 6)}^{3} x - 3 (x^{2} - 2) ({(x^{2} + 6)}^{2} \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{6}}$

Step 3.5.2

Factor ${(x^{2} + 6)}^{2}$ out of $2 {(x^{2} + 6)}^{3} x - 3 (x^{2} - 2) ({(x^{2} + 6)}^{2} \frac{d}{d x} [x^{2} + 6])$ .

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Step 3.5.2.1

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{2} (2 (x^{2} + 6) x) - 3 (x^{2} - 2) ({(x^{2} + 6)}^{2} \frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{6}}$

Step 3.5.2.2

Factor ${(x^{2} + 6)}^{2}$ out of $- 3 (x^{2} - 2) ({(x^{2} + 6)}^{2} \frac{d}{d x} [x^{2} + 6])$ .

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{2} (2 (x^{2} + 6) x) + {(x^{2} + 6)}^{2} (- 3 (x^{2} - 2) (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{6}}$

Step 3.5.2.3

Factor ${(x^{2} + 6)}^{2}$ out of ${(x^{2} + 6)}^{2} (2 (x^{2} + 6) x) + {(x^{2} + 6)}^{2} (- 3 (x^{2} - 2) (\frac{d}{d x} [x^{2} + 6]))$ .

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{2} (2 (x^{2} + 6) x - 3 (x^{2} - 2) (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{6}}$

Step 3.6

Cancel the common factors.

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Step 3.6.1

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

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{2} (2 (x^{2} + 6) x - 3 (x^{2} - 2) (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{2} {(x^{2} + 6)}^{4}}$

Step 3.6.2

Cancel the common factor.

$f''' (x) = - 36 \frac{{(x^{2} + 6)}^{2} (2 (x^{2} + 6) x - 3 (x^{2} - 2) (\frac{d}{d x} (x^{2} + 6)))}{{(x^{2} + 6)}^{2} {(x^{2} + 6)}^{4}}$

Step 3.6.3

Rewrite the expression.

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 3 (x^{2} - 2) (\frac{d}{d x} (x^{2} + 6))}{{(x^{2} + 6)}^{4}}$

Step 3.7

By the Sum Rule, the derivative of $x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]$ .

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 3 (x^{2} - 2) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{4}}$

Step 3.8

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

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 3 (x^{2} - 2) (2 x + \frac{d}{d x} (6))}{{(x^{2} + 6)}^{4}}$

Step 3.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 3 (x^{2} - 2) (2 x + 0)}{{(x^{2} + 6)}^{4}}$

Step 3.10

Combine fractions.

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Step 3.10.1

Add $2 x$ and $0$ .

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 3 (x^{2} - 2) (2 x)}{{(x^{2} + 6)}^{4}}$

Step 3.10.2

Multiply $2$ by $- 3$ .

$f''' (x) = - 36 \frac{2 (x^{2} + 6) x - 6 (x^{2} - 2) x}{{(x^{2} + 6)}^{4}}$

Step 3.10.3

Combine $- 36$ and $\frac{2 (x^{2} + 6) x - 6 (x^{2} - 2) x}{{(x^{2} + 6)}^{4}}$ .

$f''' (x) = \frac{- 36 (2 (x^{2} + 6) x - 6 (x^{2} - 2) x)}{{(x^{2} + 6)}^{4}}$

Step 3.10.4

Move the negative in front of the fraction.

$f''' (x) = - \frac{(36) (2 (x^{2} + 6) x - 6 (x^{2} - 2) x)}{{(x^{2} + 6)}^{4}}$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$f''' (x) = - \frac{36 ((2 x^{2} + 2 \cdot 6) x - 6 (x^{2} - 2) x)}{{(x^{2} + 6)}^{4}}$

Step 3.11.2

Apply the distributive property.

$f''' (x) = - \frac{36 (2 x^{2} x + 2 \cdot (6 x) - 6 (x^{2} - 2) x)}{{(x^{2} + 6)}^{4}}$

Step 3.11.3

Apply the distributive property.

$f''' (x) = - \frac{36 (2 x^{2} x + 2 \cdot (6 x) + (- 6 x^{2} - 6 \cdot - 2) x)}{{(x^{2} + 6)}^{4}}$

Step 3.11.4

Apply the distributive property.

$f''' (x) = - \frac{36 (2 x^{2} x + 2 \cdot (6 x) - 6 x^{2} x - 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.5

Apply the distributive property.

$f''' (x) = - \frac{36 (2 x^{2} x) + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6

Simplify the numerator.

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Step 3.11.6.1

Simplify each term.

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Step 3.11.6.1.1

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

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Step 3.11.6.1.1.1

Move $x$ .

$f''' (x) = - \frac{36 (2 (x \cdot x^{2})) + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.1.2

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

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Step 3.11.6.1.1.2.1

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

$f''' (x) = - \frac{36 (2 (x \cdot x^{2})) + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.1.2.2

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

$f''' (x) = - \frac{36 (2 x^{1 + 2}) + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.1.3

Add $1$ and $2$ .

$f''' (x) = - \frac{36 (2 x^{3}) + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.2

Multiply $2$ by $36$ .

$f''' (x) = - \frac{72 x^{3} + 36 (2 \cdot (6 x)) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.3

Multiply $2$ by $6$ .

$f''' (x) = - \frac{72 x^{3} + 36 (12 x) + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.4

Multiply $12$ by $36$ .

$f''' (x) = - \frac{72 x^{3} + 432 x + 36 (- 6 x^{2} x) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.5

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

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Step 3.11.6.1.5.1

Move $x$ .

$f''' (x) = - \frac{72 x^{3} + 432 x + 36 (- 6 (x \cdot x^{2})) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.5.2

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

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Step 3.11.6.1.5.2.1

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

$f''' (x) = - \frac{72 x^{3} + 432 x + 36 (- 6 (x \cdot x^{2})) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.5.2.2

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

$f''' (x) = - \frac{72 x^{3} + 432 x + 36 (- 6 x^{1 + 2}) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.5.3

Add $1$ and $2$ .

$f''' (x) = - \frac{72 x^{3} + 432 x + 36 (- 6 x^{3}) + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.6

Multiply $- 6$ by $36$ .

$f''' (x) = - \frac{72 x^{3} + 432 x - 216 x^{3} + 36 (- 6 \cdot (- 2 x))}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.7

Multiply $- 6$ by $- 2$ .

$f''' (x) = - \frac{72 x^{3} + 432 x - 216 x^{3} + 36 (12 x)}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.1.8

Multiply $12$ by $36$ .

$f''' (x) = - \frac{72 x^{3} + 432 x - 216 x^{3} + 432 x}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.2

Subtract $216 x^{3}$ from $72 x^{3}$ .

$f''' (x) = - \frac{- 144 x^{3} + 432 x + 432 x}{{(x^{2} + 6)}^{4}}$

Step 3.11.6.3

Add $432 x$ and $432 x$ .

$f''' (x) = - \frac{- 144 x^{3} + 864 x}{{(x^{2} + 6)}^{4}}$

Step 3.11.7

Factor $144 x$ out of $- 144 x^{3} + 864 x$ .

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Step 3.11.7.1

Factor $144 x$ out of $- 144 x^{3}$ .

$f''' (x) = - \frac{144 x (- x^{2}) + 864 x}{{(x^{2} + 6)}^{4}}$

Step 3.11.7.2

Factor $144 x$ out of $864 x$ .

$f''' (x) = - \frac{144 x (- x^{2}) + 144 x (6)}{{(x^{2} + 6)}^{4}}$

Step 3.11.7.3

Factor $144 x$ out of $144 x (- x^{2}) + 144 x (6)$ .

$f''' (x) = - \frac{144 x (- x^{2} + 6)}{{(x^{2} + 6)}^{4}}$

Step 3.11.8

Factor $- 1$ out of $- x^{2}$ .

$f''' (x) = - \frac{144 x (- (x^{2}) + 6)}{{(x^{2} + 6)}^{4}}$

Step 3.11.9

Rewrite $6$ as $- 1 (- 6)$ .

$f''' (x) = - \frac{144 x (- (x^{2}) - 1 \cdot - 6)}{{(x^{2} + 6)}^{4}}$

Step 3.11.10

Factor $- 1$ out of $- (x^{2}) - 1 (- 6)$ .

$f''' (x) = - \frac{144 x (- (x^{2} - 6))}{{(x^{2} + 6)}^{4}}$

Step 3.11.11

Rewrite $- (x^{2} - 6)$ as $- 1 (x^{2} - 6)$ .

$f''' (x) = - \frac{144 x (- 1 (x^{2} - 6))}{{(x^{2} + 6)}^{4}}$

Step 3.11.12

Move the negative in front of the fraction.

$f''' (x) = \frac{(144 x) (x^{2} - 6)}{{(x^{2} + 6)}^{4}}$

Step 3.11.13

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{(144 x) (x^{2} - 6)}{{(x^{2} + 6)}^{4}})$

Step 3.11.14

Multiply $\frac{(144 x) (x^{2} - 6)}{{(x^{2} + 6)}^{4}}$ by $1$ .

$f''' (x) = \frac{(144 x) (x^{2} - 6)}{{(x^{2} + 6)}^{4}}$