Calculus Examples

$f (x) = {(x^{2} + 8)}^{9}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = x^{2} + 8$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} + 8$ .

$f' (x) = \frac{d}{d u} (u^{9}) \frac{d}{d x} (x^{2} + 8)$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$f' (x) = 9 u^{8} \frac{d}{d x} (x^{2} + 8)$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} + 8$ .

$f' (x) = 9 {(x^{2} + 8)}^{8} \frac{d}{d x} (x^{2} + 8)$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$f' (x) = 9 {(x^{2} + 8)}^{8} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (8))$

Step 1.2.2

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

$f' (x) = 9 {(x^{2} + 8)}^{8} (2 x + \frac{d}{d x} (8))$

Step 1.2.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f' (x) = 9 {(x^{2} + 8)}^{8} (2 x + 0)$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$f' (x) = 9 {(x^{2} + 8)}^{8} (2 x)$

Step 1.2.4.2

Multiply $2$ by $9$ .

$f' (x) = 18 {(x^{2} + 8)}^{8} x$

Step 1.2.4.3

Reorder the factors of $18 {(x^{2} + 8)}^{8} x$ .

$f' (x) = 18 x {(x^{2} + 8)}^{8}$

Step 2

Find the second derivative.

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Step 2.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x {(x^{2} + 8)}^{8}$ with respect to $x$ is $18 \frac{d}{d x} [x {(x^{2} + 8)}^{8}]$ .

$f'' (x) = 18 \frac{d}{d x} (x {(x^{2} + 8)}^{8})$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x^{2} + 8)}^{8}$ .

$f'' (x) = 18 (x \frac{d}{d x} ({(x^{2} + 8)}^{8}) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{8}$ and $g (x) = x^{2} + 8$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2} + 8$ .

$f'' (x) = 18 (x (\frac{d}{d u} (u^{8}) \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$f'' (x) = 18 (x (8 u^{7} \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.3.3

Replace all occurrences of $u$ with $x^{2} + 8$ .

$f'' (x) = 18 (x (8 {(x^{2} + 8)}^{7} \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$f'' (x) = 18 (x (8 {(x^{2} + 8)}^{7} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (8))) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.4.2

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

$f'' (x) = 18 (x (8 {(x^{2} + 8)}^{7} (2 x + \frac{d}{d x} (8))) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.4.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = 18 (x (8 {(x^{2} + 8)}^{7} (2 x + 0)) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $2 x$ and $0$ .

$f'' (x) = 18 (x (8 {(x^{2} + 8)}^{7} (2 x)) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.4.4.2

Multiply $2$ by $8$ .

$f'' (x) = 18 (x (16 {(x^{2} + 8)}^{7} x) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.5

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

$f'' (x) = 18 (x \cdot x (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.6

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

$f'' (x) = 18 (x \cdot x (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.7

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

$f'' (x) = 18 (x^{1 + 1} (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.8

Add $1$ and $1$ .

$f'' (x) = 18 (x^{2} (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8} \frac{d}{d x} (x))$

Step 2.9

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

$f'' (x) = 18 (x^{2} (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8} \cdot 1)$

Step 2.10

Multiply ${(x^{2} + 8)}^{8}$ by $1$ .

$f'' (x) = 18 (x^{2} (16 {(x^{2} + 8)}^{7}) + {(x^{2} + 8)}^{8})$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$f'' (x) = 18 (x^{2} (16 {(x^{2} + 8)}^{7})) + 18 {(x^{2} + 8)}^{8}$

Step 2.11.2

Multiply $16$ by $18$ .

$f'' (x) = 288 (x^{2} ({(x^{2} + 8)}^{7})) + 18 {(x^{2} + 8)}^{8}$

Step 2.11.3

Factor $18 {(x^{2} + 8)}^{7}$ out of $288 x^{2} {(x^{2} + 8)}^{7} + 18 {(x^{2} + 8)}^{8}$ .

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Step 2.11.3.1

Factor $18 {(x^{2} + 8)}^{7}$ out of $288 x^{2} {(x^{2} + 8)}^{7}$ .

$f'' (x) = 18 {(x^{2} + 8)}^{7} (16 x^{2}) + 18 {(x^{2} + 8)}^{8}$

Step 2.11.3.2

Factor $18 {(x^{2} + 8)}^{7}$ out of $18 {(x^{2} + 8)}^{8}$ .

$f'' (x) = 18 {(x^{2} + 8)}^{7} (16 x^{2}) + 18 {(x^{2} + 8)}^{7} (x^{2} + 8)$

Step 2.11.3.3

Factor $18 {(x^{2} + 8)}^{7}$ out of $18 {(x^{2} + 8)}^{7} (16 x^{2}) + 18 {(x^{2} + 8)}^{7} (x^{2} + 8)$ .

$f'' (x) = 18 {(x^{2} + 8)}^{7} (16 x^{2} + x^{2} + 8)$

Step 2.11.4

Add $16 x^{2}$ and $x^{2}$ .

$f'' (x) = 18 {(x^{2} + 8)}^{7} (17 x^{2} + 8)$

Step 3

Find the third derivative.

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Step 3.1

Since $18$ is constant with respect to $x$ , the derivative of $18 {(x^{2} + 8)}^{7} (17 x^{2} + 8)$ with respect to $x$ is $18 \frac{d}{d x} [{(x^{2} + 8)}^{7} (17 x^{2} + 8)]$ .

$f''' (x) = 18 \frac{d}{d x} ({(x^{2} + 8)}^{7} (17 x^{2} + 8))$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(x^{2} + 8)}^{7}$ and $g (x) = 17 x^{2} + 8$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} \frac{d}{d x} (17 x^{2} + 8) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $17 x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [17 x^{2}] + \frac{d}{d x} [8]$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (\frac{d}{d x} (17 x^{2}) + \frac{d}{d x} (8)) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.2

Since $17$ is constant with respect to $x$ , the derivative of $17 x^{2}$ with respect to $x$ is $17 \frac{d}{d x} [x^{2}]$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (17 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8)) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.3

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

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (17 (2 x) + \frac{d}{d x} (8)) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.4

Multiply $2$ by $17$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (34 x + \frac{d}{d x} (8)) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (34 x + 0) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $34 x$ and $0$ .

$f''' (x) = 18 ({(x^{2} + 8)}^{7} (34 x) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

Step 3.3.6.2

Move $34$ to the left of ${(x^{2} + 8)}^{7}$ .

$f''' (x) = 18 (34 \cdot ({(x^{2} + 8)}^{7} x) + (17 x^{2} + 8) \frac{d}{d x} ({(x^{2} + 8)}^{7}))$

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 $f (x) = x^{7}$ and $g (x) = x^{2} + 8$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 8$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + (17 x^{2} + 8) (\frac{d}{d u} (u^{7}) \frac{d}{d x} (x^{2} + 8)))$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + (17 x^{2} + 8) (7 u^{6} \frac{d}{d x} (x^{2} + 8)))$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} + 8$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + (17 x^{2} + 8) (7 {(x^{2} + 8)}^{6} \frac{d}{d x} (x^{2} + 8)))$

Step 3.5

Differentiate.

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Step 3.5.1

Move $7$ to the left of $17 x^{2} + 8$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 7 \cdot ((17 x^{2} + 8) {(x^{2} + 8)}^{6}) \frac{d}{d x} (x^{2} + 8))$

Step 3.5.2

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 7 (17 x^{2} + 8) {(x^{2} + 8)}^{6} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (8)))$

Step 3.5.3

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

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 7 (17 x^{2} + 8) {(x^{2} + 8)}^{6} (2 x + \frac{d}{d x} (8)))$

Step 3.5.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 7 (17 x^{2} + 8) {(x^{2} + 8)}^{6} (2 x + 0))$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Add $2 x$ and $0$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 7 (17 x^{2} + 8) {(x^{2} + 8)}^{6} (2 x))$

Step 3.5.5.2

Multiply $2$ by $7$ .

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + 14 (17 x^{2} + 8) {(x^{2} + 8)}^{6} x)$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x + (14 (17 x^{2}) + 14 \cdot 8) {(x^{2} + 8)}^{6} x)$

Step 3.6.2

Apply the distributive property.

$f''' (x) = 18 (34 {(x^{2} + 8)}^{7} x) + 18 ((14 (17 x^{2}) + 14 \cdot 8) {(x^{2} + 8)}^{6} x)$

Step 3.6.3

Multiply $34$ by $18$ .

$f''' (x) = 612 ({(x^{2} + 8)}^{7} x) + 18 ((14 (17 x^{2}) + 14 \cdot 8) {(x^{2} + 8)}^{6} x)$

Step 3.6.4

Multiply $17$ by $14$ .

$f''' (x) = 612 {(x^{2} + 8)}^{7} x + 18 ((238 x^{2} + 14 \cdot 8) {(x^{2} + 8)}^{6} x)$

Step 3.6.5

Multiply $14$ by $8$ .

$f''' (x) = 612 {(x^{2} + 8)}^{7} x + 18 ((238 x^{2} + 112) {(x^{2} + 8)}^{6} x)$

Step 3.6.6

Factor $18 {(x^{2} + 8)}^{6} x$ out of $612 {(x^{2} + 8)}^{7} x + 18 (238 x^{2} + 112) {(x^{2} + 8)}^{6} x$ .

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Step 3.6.6.1

Factor $18 {(x^{2} + 8)}^{6} x$ out of $612 {(x^{2} + 8)}^{7} x$ .

$f''' (x) = 18 {(x^{2} + 8)}^{6} x (34 (x^{2} + 8)) + 18 (238 x^{2} + 112) {(x^{2} + 8)}^{6} x$

Step 3.6.6.2

Factor $18 {(x^{2} + 8)}^{6} x$ out of $18 (238 x^{2} + 112) {(x^{2} + 8)}^{6} x$ .

$f''' (x) = 18 {(x^{2} + 8)}^{6} x (34 (x^{2} + 8)) + 18 {(x^{2} + 8)}^{6} x (238 x^{2} + 112)$

Step 3.6.6.3

Factor $18 {(x^{2} + 8)}^{6} x$ out of $18 {(x^{2} + 8)}^{6} x (34 (x^{2} + 8)) + 18 {(x^{2} + 8)}^{6} x (238 x^{2} + 112)$ .

$f''' (x) = 18 {(x^{2} + 8)}^{6} x (34 (x^{2} + 8) + 238 x^{2} + 112)$

Step 3.6.7

Reorder the factors of $18 {(x^{2} + 8)}^{6} x (34 (x^{2} + 8) + 238 x^{2} + 112)$ .

$f''' (x) = 18 x {(x^{2} + 8)}^{6} (34 (x^{2} + 8) + 238 x^{2} + 112)$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the distributive property.

$f^{4} (x) = \frac{d}{d x} (18 x {(x^{2} + 8)}^{6} (34 x^{2} + 34 \cdot 8 + 238 x^{2} + 112))$

Step 4.1.1.2

Multiply $34$ by $8$ .

$f^{4} (x) = \frac{d}{d x} (18 x {(x^{2} + 8)}^{6} (34 x^{2} + 272 + 238 x^{2} + 112))$

Step 4.1.2

Simplify by adding terms.

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Step 4.1.2.1

Add $34 x^{2}$ and $238 x^{2}$ .

$f^{4} (x) = \frac{d}{d x} (18 x {(x^{2} + 8)}^{6} (272 x^{2} + 272 + 112))$

Step 4.1.2.2

Add $272$ and $112$ .

$f^{4} (x) = \frac{d}{d x} (18 x {(x^{2} + 8)}^{6} (272 x^{2} + 384))$

Step 4.1.3

Since $18$ is constant with respect to $x$ , the derivative of $18 x {(x^{2} + 8)}^{6} (272 x^{2} + 384)$ with respect to $x$ is $18 \frac{d}{d x} [x {(x^{2} + 8)}^{6} (272 x^{2} + 384)]$ .

$f^{4} (x) = 18 \frac{d}{d x} (x {(x^{2} + 8)}^{6} (272 x^{2} + 384))$

Step 4.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x {(x^{2} + 8)}^{6}$ and $g (x) = 272 x^{2} + 384$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} \frac{d}{d x} (272 x^{2} + 384) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3

Differentiate.

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Step 4.3.1

By the Sum Rule, the derivative of $272 x^{2} + 384$ with respect to $x$ is $\frac{d}{d x} [272 x^{2}] + \frac{d}{d x} [384]$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (\frac{d}{d x} (272 x^{2}) + \frac{d}{d x} (384)) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3.2

Since $272$ is constant with respect to $x$ , the derivative of $272 x^{2}$ with respect to $x$ is $272 \frac{d}{d x} [x^{2}]$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (272 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (384)) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3.3

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

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (272 (2 x) + \frac{d}{d x} (384)) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3.4

Multiply $2$ by $272$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (544 x + \frac{d}{d x} (384)) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3.5

Since $384$ is constant with respect to $x$ , the derivative of $384$ with respect to $x$ is $0$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (544 x + 0) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.3.6

Add $544 x$ and $0$ .

$f^{4} (x) = 18 (x {(x^{2} + 8)}^{6} (544 x) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.4

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

$f^{4} (x) = 18 (x \cdot x {(x^{2} + 8)}^{6} \cdot 544 + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.5

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

$f^{4} (x) = 18 (x \cdot x {(x^{2} + 8)}^{6} \cdot 544 + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.6

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

$f^{4} (x) = 18 (x^{1 + 1} {(x^{2} + 8)}^{6} \cdot 544 + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $1$ and $1$ .

$f^{4} (x) = 18 (x^{2} {(x^{2} + 8)}^{6} \cdot 544 + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.7.2

Move $544$ to the left of $x^{2} {(x^{2} + 8)}^{6}$ .

$f^{4} (x) = 18 (544 \cdot (x^{2} {(x^{2} + 8)}^{6}) + (272 x^{2} + 384) \frac{d}{d x} (x {(x^{2} + 8)}^{6}))$

Step 4.8

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 $g (x) = {(x^{2} + 8)}^{6}$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x \frac{d}{d x} ({(x^{2} + 8)}^{6}) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{2} + 8$ .

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Step 4.9.1

To apply the Chain Rule, set $u$ as $x^{2} + 8$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (\frac{d}{d u} (u^{6}) \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.9.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 u^{5} \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.9.3

Replace all occurrences of $u$ with $x^{2} + 8$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 {(x^{2} + 8)}^{5} \frac{d}{d x} (x^{2} + 8)) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.10

Differentiate.

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Step 4.10.1

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 {(x^{2} + 8)}^{5} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (8))) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.10.2

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

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 {(x^{2} + 8)}^{5} (2 x + \frac{d}{d x} (8))) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.10.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 {(x^{2} + 8)}^{5} (2 x + 0)) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.10.4

Simplify the expression.

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Step 4.10.4.1

Add $2 x$ and $0$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (6 {(x^{2} + 8)}^{5} (2 x)) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.10.4.2

Multiply $2$ by $6$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x (12 {(x^{2} + 8)}^{5} x) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.11

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

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x \cdot x (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.12

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

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x \cdot x (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.13

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

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{1 + 1} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.14

Add $1$ and $1$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{2} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6} \frac{d}{d x} (x)))$

Step 4.15

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

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{2} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6} \cdot 1))$

Step 4.16

Multiply ${(x^{2} + 8)}^{6}$ by $1$ .

$f^{4} (x) = 18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{2} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6}))$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{2} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6}))$ .

$18 (544 x^{2} {(x^{2} + 8)}^{6} + (272 x^{2} + 384) (x^{2} (12 {(x^{2} + 8)}^{5}) + {(x^{2} + 8)}^{6}))$