Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{2} + 3}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{2} + 3}]$ .

$6 \frac{d}{d x} [\frac{1}{x^{2} + 3}]$

Step 1.1.2

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

$6 \frac{d}{d x} [{(x^{2} + 3)}^{- 1}]$

Step 1.2

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

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Step 1.2.1

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

$6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 3])$

Step 1.2.2

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

$6 (- u^{- 2} \frac{d}{d x} [x^{2} + 3])$

Step 1.2.3

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

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

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $6$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 6 {(x^{2} + 3)}^{- 2} (2 x + 0)$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $2 x$ and $0$ .

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

Step 1.3.5.2

Multiply $2$ by $- 6$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $- 12$ and $\frac{1}{{(x^{2} + 3)}^{2}}$ .

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

Step 1.4.2.2

Move the negative in front of the fraction.

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

Step 1.4.2.3

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

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

Step 1.4.2.4

Move $12$ to the left of $x$ .

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

Step 2

Find the second derivative.

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Step 2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- \frac{12 x}{{(x^{2} + 3)}^{2}}$ with respect to $x$ is $- 12 \frac{d}{d x} [\frac{x}{{(x^{2} + 3)}^{2}}]$ .

$- 12 \frac{d}{d x} [\frac{x}{{(x^{2} + 3)}^{2}}]$

Step 2.2

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

$- 12 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 3)}^{2}]}{{({(x^{2} + 3)}^{2})}^{2}}$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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Step 2.3.1.1

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

$- 12 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 3)}^{2}]}{{(x^{2} + 3)}^{2 \cdot 2}}$

Step 2.3.1.2

Multiply $2$ by $2$ .

$- 12 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 3)}^{2}]}{{(x^{2} + 3)}^{4}}$

Step 2.3.2

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

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

Step 2.3.3

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

$- 12 \frac{{(x^{2} + 3)}^{2} - x \frac{d}{d x} [{(x^{2} + 3)}^{2}]}{{(x^{2} + 3)}^{4}}$

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

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Step 2.4.1

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

$- 12 \frac{{(x^{2} + 3)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 2.4.2

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

$- 12 \frac{{(x^{2} + 3)}^{2} - x (2 u \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 2.4.3

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

$- 12 \frac{{(x^{2} + 3)}^{2} - x (2 (x^{2} + 3) \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

$- 12 \frac{{(x^{2} + 3)}^{2} - 2 x ((x^{2} + 3) \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 2.5.2

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

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Step 2.5.2.1

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

$- 12 \frac{(x^{2} + 3) (x^{2} + 3) - 2 x ((x^{2} + 3) \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 2.5.2.2

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

$- 12 \frac{(x^{2} + 3) (x^{2} + 3) + (x^{2} + 3) (- 2 x (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{4}}$

Step 2.5.2.3

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

$- 12 \frac{(x^{2} + 3) (x^{2} + 3 - 2 x (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{4}}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$- 12 \frac{(x^{2} + 3) (x^{2} + 3 - 2 x (\frac{d}{d x} [x^{2} + 3]))}{(x^{2} + 3) {(x^{2} + 3)}^{3}}$

Step 2.6.2

Cancel the common factor.

$- 12 \frac{(x^{2} + 3) (x^{2} + 3 - 2 x (\frac{d}{d x} [x^{2} + 3]))}{(x^{2} + 3) {(x^{2} + 3)}^{3}}$

Step 2.6.3

Rewrite the expression.

$- 12 \frac{x^{2} + 3 - 2 x (\frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{3}}$

Step 2.7

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

$- 12 \frac{x^{2} + 3 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{3}}$

Step 2.8

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

$- 12 \frac{x^{2} + 3 - 2 x (2 x + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{3}}$

Step 2.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 12 \frac{x^{2} + 3 - 2 x (2 x + 0)}{{(x^{2} + 3)}^{3}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 2$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Add $1$ and $1$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Move the negative in front of the fraction.

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

Step 2.18

Simplify.

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Step 2.18.1

Apply the distributive property.

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

Step 2.18.2

Simplify each term.

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Step 2.18.2.1

Multiply $- 3$ by $12$ .

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

Step 2.18.2.2

Multiply $12$ by $3$ .

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

Step 3

Find the third derivative.

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Step 3.1

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 $f (x) = - 1$ and $g (x) = \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}}$ .

$- \frac{d}{d x} [\frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}}] + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.2

$- \frac{{(x^{2} + 3)}^{3} \frac{d}{d x} [- 36 x^{2} + 36] - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{({(x^{2} + 3)}^{3})}^{2}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3

Differentiate.

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Step 3.3.1

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

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Step 3.3.1.1

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

$- \frac{{(x^{2} + 3)}^{3} \frac{d}{d x} [- 36 x^{2} + 36] - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{3 \cdot 2}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.1.2

Multiply $3$ by $2$ .

$- \frac{{(x^{2} + 3)}^{3} \frac{d}{d x} [- 36 x^{2} + 36] - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.2

By the Sum Rule, the derivative of $- 36 x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [36]$ .

$- \frac{{(x^{2} + 3)}^{3} (\frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [36]) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.3

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

$- \frac{{(x^{2} + 3)}^{3} (- 36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.4

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

$- \frac{{(x^{2} + 3)}^{3} (- 36 (2 x) + \frac{d}{d x} [36]) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.5

Multiply $2$ by $- 36$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x + \frac{d}{d x} [36]) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.6

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x + 0) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.3.7

Add $- 72 x$ and $0$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - (- 36 x^{2} + 36) \frac{d}{d x} [{(x^{2} + 3)}^{3}]}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

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

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Step 3.4.1

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - (- 36 x^{2} + 36) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.4.2

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - (- 36 x^{2} + 36) (3 u^{2} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.4.3

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - (- 36 x^{2} + 36) (3 {(x^{2} + 3)}^{2} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5

Differentiate.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.2

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]))}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.3

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} (2 x + \frac{d}{d x} [3]))}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} (2 x + 0))}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Add $2 x$ and $0$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} (2 x))}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.5.2

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 3 (- 36 x^{2} + 36) (2 \cdot {(x^{2} + 3)}^{2} x)}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.5.3

Multiply $2$ by $- 3$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 6 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} x)}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \frac{d}{d x} [- 1]$

Step 3.5.6

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

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 6 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} x)}{{(x^{2} + 3)}^{6}} + \frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}} \cdot 0$

Step 3.5.7

Simplify the expression.

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Step 3.5.7.1

Multiply $\frac{- 36 x^{2} + 36}{{(x^{2} + 3)}^{3}}$ by $0$ .

$- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 6 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} x)}{{(x^{2} + 3)}^{6}} + 0$

Step 3.5.7.2

Add $- \frac{{(x^{2} + 3)}^{3} (- 72 x) - 6 (- 36 x^{2} + 36) ({(x^{2} + 3)}^{2} x)}{{(x^{2} + 3)}^{6}}$ and $0$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Simplify the numerator.

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Step 3.6.2.1

Factor ${(x^{2} + 3)}^{2} x$ out of ${(x^{2} + 3)}^{3} (- 72 x) + (- 6 (- 36 x^{2}) - 6 \cdot 36) {(x^{2} + 3)}^{2} x$ .

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Step 3.6.2.1.1

Factor ${(x^{2} + 3)}^{2} x$ out of ${(x^{2} + 3)}^{3} (- 72 x)$ .

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

Step 3.6.2.1.2

Factor ${(x^{2} + 3)}^{2} x$ out of $(- 6 (- 36 x^{2}) - 6 \cdot 36) {(x^{2} + 3)}^{2} x$ .

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

Step 3.6.2.1.3

Factor ${(x^{2} + 3)}^{2} x$ out of ${(x^{2} + 3)}^{2} x ((x^{2} + 3) \cdot (- 72)) + {(x^{2} + 3)}^{2} x (- 6 (- 36 x^{2}) - 6 \cdot 36)$ .

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

Step 3.6.2.2

Combine exponents.

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Step 3.6.2.2.1

Multiply $- 36$ by $- 6$ .

$- \frac{{(x^{2} + 3)}^{2} x ((x^{2} + 3) \cdot (- 72) + 216 x^{2} - 6 \cdot 36)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.2.2

Multiply $- 6$ by $36$ .

$- \frac{{(x^{2} + 3)}^{2} x ((x^{2} + 3) \cdot (- 72) + 216 x^{2} - 216)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.3

Simplify each term.

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Step 3.6.2.3.1

Apply the distributive property.

$- \frac{{(x^{2} + 3)}^{2} x (x^{2} \cdot - 72 + 3 \cdot - 72 + 216 x^{2} - 216)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.3.2

Move $- 72$ to the left of $x^{2}$ .

$- \frac{{(x^{2} + 3)}^{2} x (- 72 \cdot x^{2} + 3 \cdot - 72 + 216 x^{2} - 216)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.3.3

Multiply $3$ by $- 72$ .

$- \frac{{(x^{2} + 3)}^{2} x (- 72 x^{2} - 216 + 216 x^{2} - 216)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.4

Add $- 72 x^{2}$ and $216 x^{2}$ .

$- \frac{{(x^{2} + 3)}^{2} x (144 x^{2} - 216 - 216)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.5

Subtract $216$ from $- 216$ .

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

Step 3.6.2.6

Factor $144$ out of $144 x^{2} - 432$ .

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Step 3.6.2.6.1

Factor $144$ out of $144 x^{2}$ .

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

Step 3.6.2.6.2

Factor $144$ out of $- 432$ .

$- \frac{{(x^{2} + 3)}^{2} x (144 x^{2} + 144 \cdot - 3)}{{(x^{2} + 3)}^{6}}$

Step 3.6.2.6.3

Factor $144$ out of $144 x^{2} + 144 \cdot - 3$ .

$- \frac{{(x^{2} + 3)}^{2} x (144 (x^{2} - 3))}{{(x^{2} + 3)}^{6}}$

$- \frac{{(x^{2} + 3)}^{2} x \cdot 144 (x^{2} - 3)}{{(x^{2} + 3)}^{6}}$

Step 3.6.3

Combine terms.

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Step 3.6.3.1

Move $144$ to the left of ${(x^{2} + 3)}^{2} x$ .

$- \frac{144 \cdot ({(x^{2} + 3)}^{2} x) (x^{2} - 3)}{{(x^{2} + 3)}^{6}}$

Step 3.6.3.2

Cancel the common factor of ${(x^{2} + 3)}^{2}$ and ${(x^{2} + 3)}^{6}$ .

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Step 3.6.3.2.1

Factor ${(x^{2} + 3)}^{2}$ out of $144 {(x^{2} + 3)}^{2} x (x^{2} - 3)$ .

$- \frac{{(x^{2} + 3)}^{2} (144 x (x^{2} - 3))}{{(x^{2} + 3)}^{6}}$

Step 3.6.3.2.2

Cancel the common factors.

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Step 3.6.3.2.2.1

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

$- \frac{{(x^{2} + 3)}^{2} (144 x (x^{2} - 3))}{{(x^{2} + 3)}^{2} {(x^{2} + 3)}^{4}}$

Step 3.6.3.2.2.2

Cancel the common factor.

$- \frac{{(x^{2} + 3)}^{2} (144 x (x^{2} - 3))}{{(x^{2} + 3)}^{2} {(x^{2} + 3)}^{4}}$

Step 3.6.3.2.2.3

Rewrite the expression.

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

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 144$ is constant with respect to $x$ , the derivative of $- \frac{144 x (x^{2} - 3)}{{(x^{2} + 3)}^{4}}$ with respect to $x$ is $- 144 \frac{d}{d x} [\frac{x (x^{2} - 3)}{{(x^{2} + 3)}^{4}}]$ .

$- 144 \frac{d}{d x} [\frac{x (x^{2} - 3)}{{(x^{2} + 3)}^{4}}]$

Step 4.2

$- 144 \frac{{(x^{2} + 3)}^{4} \frac{d}{d x} [x (x^{2} - 3)] - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{({(x^{2} + 3)}^{4})}^{2}}$

Step 4.3

Multiply the exponents in ${({(x^{2} + 3)}^{4})}^{2}$ .

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Step 4.3.1

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

$- 144 \frac{{(x^{2} + 3)}^{4} \frac{d}{d x} [x (x^{2} - 3)] - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{4 \cdot 2}}$

Step 4.3.2

Multiply $4$ by $2$ .

$- 144 \frac{{(x^{2} + 3)}^{4} \frac{d}{d x} [x (x^{2} - 3)] - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.4

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 $g (x) = x^{2} - 3$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (x \frac{d}{d x} [x^{2} - 3] + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.5

Differentiate.

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Step 4.5.1

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

$- 144 \frac{{(x^{2} + 3)}^{4} (x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.5.2

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

$- 144 \frac{{(x^{2} + 3)}^{4} (x (2 x + \frac{d}{d x} [- 3]) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.5.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (x (2 x + 0) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.5.4

Add $2 x$ and $0$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (x (2 x) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.6

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

$- 144 \frac{{(x^{2} + 3)}^{4} (2 (x^{1} x) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.7

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

$- 144 \frac{{(x^{2} + 3)}^{4} (2 (x^{1} x^{1}) + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.8

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

$- 144 \frac{{(x^{2} + 3)}^{4} (2 x^{1 + 1} + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.9

Differentiate using the Power Rule.

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Step 4.9.1

Add $1$ and $1$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (2 x^{2} + (x^{2} - 3) \frac{d}{d x} [x]) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.9.2

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

$- 144 \frac{{(x^{2} + 3)}^{4} (2 x^{2} + (x^{2} - 3) \cdot 1) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.9.3

Simplify by adding terms.

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Step 4.9.3.1

Multiply $x^{2} - 3$ by $1$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (2 x^{2} + x^{2} - 3) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.9.3.2

Add $2 x^{2}$ and $x^{2}$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (3 x^{2} - 3) - x (x^{2} - 3) \frac{d}{d x} [{(x^{2} + 3)}^{4}]}{{(x^{2} + 3)}^{8}}$

Step 4.10

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

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Step 4.10.1

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

$- 144 \frac{{(x^{2} + 3)}^{4} (3 x^{2} - 3) - x (x^{2} - 3) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{8}}$

Step 4.10.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (3 x^{2} - 3) - x (x^{2} - 3) (4 u^{3} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{8}}$

Step 4.10.3

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

$- 144 \frac{{(x^{2} + 3)}^{4} (3 x^{2} - 3) - x (x^{2} - 3) (4 {(x^{2} + 3)}^{3} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{8}}$

Step 4.11

Simplify with factoring out.

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Step 4.11.1

Multiply $4$ by $- 1$ .

$- 144 \frac{{(x^{2} + 3)}^{4} (3 x^{2} - 3) - 4 x (x^{2} - 3) ({(x^{2} + 3)}^{3} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{8}}$

Step 4.11.2

Factor ${(x^{2} + 3)}^{3}$ out of ${(x^{2} + 3)}^{4} (3 x^{2} - 3) - 4 x (x^{2} - 3) ({(x^{2} + 3)}^{3} \frac{d}{d x} [x^{2} + 3])$ .

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Step 4.11.2.1

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

$- 144 \frac{{(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3)) - 4 x (x^{2} - 3) ({(x^{2} + 3)}^{3} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{8}}$

Step 4.11.2.2

Factor ${(x^{2} + 3)}^{3}$ out of $- 4 x (x^{2} - 3) ({(x^{2} + 3)}^{3} \frac{d}{d x} [x^{2} + 3])$ .

$- 144 \frac{{(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3)) + {(x^{2} + 3)}^{3} (- 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{8}}$

Step 4.11.2.3

Factor ${(x^{2} + 3)}^{3}$ out of ${(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3)) + {(x^{2} + 3)}^{3} (- 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3]))$ .

$- 144 \frac{{(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{8}}$

Step 4.12

Cancel the common factors.

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Step 4.12.1

Factor ${(x^{2} + 3)}^{3}$ out of ${(x^{2} + 3)}^{8}$ .

$- 144 \frac{{(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{3} {(x^{2} + 3)}^{5}}$

Step 4.12.2

Cancel the common factor.

$- 144 \frac{{(x^{2} + 3)}^{3} ((x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3]))}{{(x^{2} + 3)}^{3} {(x^{2} + 3)}^{5}}$

Step 4.12.3

Rewrite the expression.

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{5}}$

Step 4.13

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

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{5}}$

Step 4.14

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

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (2 x + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{5}}$

Step 4.15

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (2 x + 0)}{{(x^{2} + 3)}^{5}}$

Step 4.16

Simplify the expression.

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Step 4.16.1

Add $2 x$ and $0$ .

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 4 x (x^{2} - 3) (2 x)}{{(x^{2} + 3)}^{5}}$

Step 4.16.2

Multiply $2$ by $- 4$ .

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 8 x (x^{2} - 3) x}{{(x^{2} + 3)}^{5}}$

Step 4.17

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

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 8 (x^{1} x) (x^{2} - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.18

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

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 8 (x^{1} x^{1}) (x^{2} - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.19

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

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 8 x^{1 + 1} (x^{2} - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.20

Add $1$ and $1$ .

$- 144 \frac{(x^{2} + 3) (3 x^{2} - 3) - 8 x^{2} (x^{2} - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.21

Combine $- 144$ and $\frac{(x^{2} + 3) (3 x^{2} - 3) - 8 x^{2} (x^{2} - 3)}{{(x^{2} + 3)}^{5}}$ .

$\frac{- 144 ((x^{2} + 3) (3 x^{2} - 3) - 8 x^{2} (x^{2} - 3))}{{(x^{2} + 3)}^{5}}$

Step 4.22

Move the negative in front of the fraction.

$- \frac{144 ((x^{2} + 3) (3 x^{2} - 3) - 8 x^{2} (x^{2} - 3))}{{(x^{2} + 3)}^{5}}$

Step 4.23

Simplify.

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Step 4.23.1

Apply the distributive property.

$- \frac{144 ((x^{2} + 3) (3 x^{2} - 3) - 8 x^{2} x^{2} - 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.2

Apply the distributive property.

$- \frac{144 ((x^{2} + 3) (3 x^{2} - 3)) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3

Simplify the numerator.

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Step 4.23.3.1

Simplify each term.

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Step 4.23.3.1.1

Expand $(x^{2} + 3) (3 x^{2} - 3)$ using the FOIL Method.

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Step 4.23.3.1.1.1

Apply the distributive property.

$- \frac{144 (x^{2} (3 x^{2} - 3) + 3 (3 x^{2} - 3)) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.1.2

Apply the distributive property.

$- \frac{144 (x^{2} (3 x^{2}) + x^{2} \cdot - 3 + 3 (3 x^{2} - 3)) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.1.3

Apply the distributive property.

$- \frac{144 (x^{2} (3 x^{2}) + x^{2} \cdot - 3 + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2

Simplify and combine like terms.

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Step 4.23.3.1.2.1

Simplify each term.

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Step 4.23.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$- \frac{144 (3 x^{2} x^{2} + x^{2} \cdot - 3 + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.2

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

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Step 4.23.3.1.2.1.2.1

Move $x^{2}$ .

$- \frac{144 (3 (x^{2} x^{2}) + x^{2} \cdot - 3 + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.2.2

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

$- \frac{144 (3 x^{2 + 2} + x^{2} \cdot - 3 + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.2.3

Add $2$ and $2$ .

$- \frac{144 (3 x^{4} + x^{2} \cdot - 3 + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.3

Move $- 3$ to the left of $x^{2}$ .

$- \frac{144 (3 x^{4} - 3 \cdot x^{2} + 3 (3 x^{2}) + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.4

Multiply $3$ by $3$ .

$- \frac{144 (3 x^{4} - 3 x^{2} + 9 x^{2} + 3 \cdot - 3) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.1.5

Multiply $3$ by $- 3$ .

$- \frac{144 (3 x^{4} - 3 x^{2} + 9 x^{2} - 9) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.2.2

Add $- 3 x^{2}$ and $9 x^{2}$ .

$- \frac{144 (3 x^{4} + 6 x^{2} - 9) + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.3

Apply the distributive property.

$- \frac{144 (3 x^{4}) + 144 (6 x^{2}) + 144 \cdot - 9 + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.4

Simplify.

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Step 4.23.3.1.4.1

Multiply $3$ by $144$ .

$- \frac{432 x^{4} + 144 (6 x^{2}) + 144 \cdot - 9 + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.4.2

Multiply $6$ by $144$ .

$- \frac{432 x^{4} + 864 x^{2} + 144 \cdot - 9 + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.4.3

Multiply $144$ by $- 9$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 + 144 (- 8 x^{2} x^{2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.5

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

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Step 4.23.3.1.5.1

Move $x^{2}$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 + 144 (- 8 (x^{2} x^{2})) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.5.2

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

$- \frac{432 x^{4} + 864 x^{2} - 1296 + 144 (- 8 x^{2 + 2}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.5.3

Add $2$ and $2$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 + 144 (- 8 x^{4}) + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.6

Multiply $- 8$ by $144$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 - 1152 x^{4} + 144 (- 8 x^{2} \cdot - 3)}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.7

Multiply $- 3$ by $- 8$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 - 1152 x^{4} + 144 (24 x^{2})}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.1.8

Multiply $24$ by $144$ .

$- \frac{432 x^{4} + 864 x^{2} - 1296 - 1152 x^{4} + 3456 x^{2}}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.2

Subtract $1152 x^{4}$ from $432 x^{4}$ .

$- \frac{- 720 x^{4} + 864 x^{2} - 1296 + 3456 x^{2}}{{(x^{2} + 3)}^{5}}$

Step 4.23.3.3

Add $864 x^{2}$ and $3456 x^{2}$ .

$- \frac{- 720 x^{4} + 4320 x^{2} - 1296}{{(x^{2} + 3)}^{5}}$

Step 4.23.4

Factor $144$ out of $- 720 x^{4} + 4320 x^{2} - 1296$ .

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Step 4.23.4.1

Factor $144$ out of $- 720 x^{4}$ .

$- \frac{144 (- 5 x^{4}) + 4320 x^{2} - 1296}{{(x^{2} + 3)}^{5}}$

Step 4.23.4.2

Factor $144$ out of $4320 x^{2}$ .

$- \frac{144 (- 5 x^{4}) + 144 (30 x^{2}) - 1296}{{(x^{2} + 3)}^{5}}$

Step 4.23.4.3

Factor $144$ out of $- 1296$ .

$- \frac{144 (- 5 x^{4}) + 144 (30 x^{2}) + 144 (- 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.4.4

Factor $144$ out of $144 (- 5 x^{4}) + 144 (30 x^{2})$ .

$- \frac{144 (- 5 x^{4} + 30 x^{2}) + 144 (- 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.4.5

Factor $144$ out of $144 (- 5 x^{4} + 30 x^{2}) + 144 (- 9)$ .

$- \frac{144 (- 5 x^{4} + 30 x^{2} - 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.5

Factor $- 1$ out of $- 5 x^{4}$ .

$- \frac{144 (- (5 x^{4}) + 30 x^{2} - 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.6

Factor $- 1$ out of $30 x^{2}$ .

$- \frac{144 (- (5 x^{4}) - (- 30 x^{2}) - 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.7

Factor $- 1$ out of $- (5 x^{4}) - (- 30 x^{2})$ .

$- \frac{144 (- (5 x^{4} - 30 x^{2}) - 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.8

Rewrite $- 9$ as $- 1 (9)$ .

$- \frac{144 (- (5 x^{4} - 30 x^{2}) - 1 (9))}{{(x^{2} + 3)}^{5}}$

Step 4.23.9

Factor $- 1$ out of $- (5 x^{4} - 30 x^{2}) - 1 (9)$ .

$- \frac{144 (- (5 x^{4} - 30 x^{2} + 9))}{{(x^{2} + 3)}^{5}}$

Step 4.23.10

Rewrite $- (5 x^{4} - 30 x^{2} + 9)$ as $- 1 (5 x^{4} - 30 x^{2} + 9)$ .

$- \frac{144 (- 1 (5 x^{4} - 30 x^{2} + 9))}{{(x^{2} + 3)}^{5}}$

Step 4.23.11

Move the negative in front of the fraction.

$- - \frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.12

Multiply $- 1$ by $- 1$ .

$1 \frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$

Step 4.23.13

Multiply $\frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$ by $1$ .

$f^{4} (x) = \frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$ .

$\frac{144 (5 x^{4} - 30 x^{2} + 9)}{{(x^{2} + 3)}^{5}}$