Calculus Examples

$12 x {(x^{2} + 10)}^{5}$

Step 1

Find the first derivative.

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Step 1.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x {(x^{2} + 10)}^{5}$ with respect to $x$ is $12 \frac{d}{d x} [x {(x^{2} + 10)}^{5}]$ .

$12 \frac{d}{d x} [x {(x^{2} + 10)}^{5}]$

Step 1.2

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 $g (x) = {(x^{2} + 10)}^{5}$ .

$12 (x \frac{d}{d x} [{(x^{2} + 10)}^{5}] + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} + 10$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x^{2} + 10$ .

$12 (x (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$12 (x (5 u^{4} \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.3.3

Replace all occurrences of $u$ with $x^{2} + 10$ .

$12 (x (5 {(x^{2} + 10)}^{4} \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $x^{2} + 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10]$ .

$12 (x (5 {(x^{2} + 10)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10])) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.4.2

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

$12 (x (5 {(x^{2} + 10)}^{4} (2 x + \frac{d}{d x} [10])) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.4.3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$12 (x (5 {(x^{2} + 10)}^{4} (2 x + 0)) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Add $2 x$ and $0$ .

$12 (x (5 {(x^{2} + 10)}^{4} (2 x)) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.4.4.2

Multiply $2$ by $5$ .

$12 (x (10 {(x^{2} + 10)}^{4} x) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.5

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

$12 (x^{1} x (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.6

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

$12 (x^{1} x^{1} (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.7

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

$12 (x^{1 + 1} (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.8

Add $1$ and $1$ .

$12 (x^{2} (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5} \frac{d}{d x} [x])$

Step 1.9

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

$12 (x^{2} (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5} \cdot 1)$

Step 1.10

Multiply ${(x^{2} + 10)}^{5}$ by $1$ .

$12 (x^{2} (10 {(x^{2} + 10)}^{4}) + {(x^{2} + 10)}^{5})$

Step 1.11

Simplify.

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Step 1.11.1

Apply the distributive property.

$12 (x^{2} (10 {(x^{2} + 10)}^{4})) + 12 {(x^{2} + 10)}^{5}$

Step 1.11.2

Multiply $10$ by $12$ .

$120 (x^{2} ({(x^{2} + 10)}^{4})) + 12 {(x^{2} + 10)}^{5}$

Step 1.11.3

Factor $12 {(x^{2} + 10)}^{4}$ out of $120 x^{2} {(x^{2} + 10)}^{4} + 12 {(x^{2} + 10)}^{5}$ .

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Step 1.11.3.1

Factor $12 {(x^{2} + 10)}^{4}$ out of $120 x^{2} {(x^{2} + 10)}^{4}$ .

$12 {(x^{2} + 10)}^{4} (10 x^{2}) + 12 {(x^{2} + 10)}^{5}$

Step 1.11.3.2

Factor $12 {(x^{2} + 10)}^{4}$ out of $12 {(x^{2} + 10)}^{5}$ .

$12 {(x^{2} + 10)}^{4} (10 x^{2}) + 12 {(x^{2} + 10)}^{4} (x^{2} + 10)$

Step 1.11.3.3

Factor $12 {(x^{2} + 10)}^{4}$ out of $12 {(x^{2} + 10)}^{4} (10 x^{2}) + 12 {(x^{2} + 10)}^{4} (x^{2} + 10)$ .

$12 {(x^{2} + 10)}^{4} (10 x^{2} + x^{2} + 10)$

Step 1.11.4

Add $10 x^{2}$ and $x^{2}$ .

$f' (x) = 12 {(x^{2} + 10)}^{4} (11 x^{2} + 10)$

Step 2

Find the second derivative.

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 {(x^{2} + 10)}^{4} (11 x^{2} + 10)$ with respect to $x$ is $12 \frac{d}{d x} [{(x^{2} + 10)}^{4} (11 x^{2} + 10)]$ .

$12 \frac{d}{d x} [{(x^{2} + 10)}^{4} (11 x^{2} + 10)]$

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 $f (x) = {(x^{2} + 10)}^{4}$ and $g (x) = 11 x^{2} + 10$ .

$12 ({(x^{2} + 10)}^{4} \frac{d}{d x} [11 x^{2} + 10] + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $11 x^{2} + 10$ with respect to $x$ is $\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [10]$ .

$12 ({(x^{2} + 10)}^{4} (\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [10]) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{2}$ with respect to $x$ is $11 \frac{d}{d x} [x^{2}]$ .

$12 ({(x^{2} + 10)}^{4} (11 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10]) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.3

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

$12 ({(x^{2} + 10)}^{4} (11 (2 x) + \frac{d}{d x} [10]) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.4

Multiply $2$ by $11$ .

$12 ({(x^{2} + 10)}^{4} (22 x + \frac{d}{d x} [10]) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.5

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$12 ({(x^{2} + 10)}^{4} (22 x + 0) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $22 x$ and $0$ .

$12 ({(x^{2} + 10)}^{4} (22 x) + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

Step 2.3.6.2

Move $22$ to the left of ${(x^{2} + 10)}^{4}$ .

$12 (22 \cdot {(x^{2} + 10)}^{4} x + (11 x^{2} + 10) \frac{d}{d x} [{(x^{2} + 10)}^{4}])$

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 $f (x) = x^{4}$ and $g (x) = x^{2} + 10$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 10$ .

$12 (22 {(x^{2} + 10)}^{4} x + (11 x^{2} + 10) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [x^{2} + 10]))$

Step 2.4.2

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

$12 (22 {(x^{2} + 10)}^{4} x + (11 x^{2} + 10) (4 u^{3} \frac{d}{d x} [x^{2} + 10]))$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 10$ .

$12 (22 {(x^{2} + 10)}^{4} x + (11 x^{2} + 10) (4 {(x^{2} + 10)}^{3} \frac{d}{d x} [x^{2} + 10]))$

Step 2.5

Differentiate.

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Step 2.5.1

Move $4$ to the left of $11 x^{2} + 10$ .

$12 (22 {(x^{2} + 10)}^{4} x + 4 \cdot (11 x^{2} + 10) {(x^{2} + 10)}^{3} \frac{d}{d x} [x^{2} + 10])$

Step 2.5.2

By the Sum Rule, the derivative of $x^{2} + 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10]$ .

$12 (22 {(x^{2} + 10)}^{4} x + 4 (11 x^{2} + 10) {(x^{2} + 10)}^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10]))$

Step 2.5.3

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

$12 (22 {(x^{2} + 10)}^{4} x + 4 (11 x^{2} + 10) {(x^{2} + 10)}^{3} (2 x + \frac{d}{d x} [10]))$

Step 2.5.4

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$12 (22 {(x^{2} + 10)}^{4} x + 4 (11 x^{2} + 10) {(x^{2} + 10)}^{3} (2 x + 0))$

Step 2.5.5

Simplify the expression.

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Step 2.5.5.1

Add $2 x$ and $0$ .

$12 (22 {(x^{2} + 10)}^{4} x + 4 (11 x^{2} + 10) {(x^{2} + 10)}^{3} (2 x))$

Step 2.5.5.2

Multiply $2$ by $4$ .

$12 (22 {(x^{2} + 10)}^{4} x + 8 (11 x^{2} + 10) {(x^{2} + 10)}^{3} x)$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$12 (22 {(x^{2} + 10)}^{4} x + (8 (11 x^{2}) + 8 \cdot 10) {(x^{2} + 10)}^{3} x)$

Step 2.6.2

Apply the distributive property.

$12 (22 {(x^{2} + 10)}^{4} x) + 12 ((8 (11 x^{2}) + 8 \cdot 10) {(x^{2} + 10)}^{3} x)$

Step 2.6.3

Multiply $22$ by $12$ .

$264 ({(x^{2} + 10)}^{4} x) + 12 ((8 (11 x^{2}) + 8 \cdot 10) {(x^{2} + 10)}^{3} x)$

Step 2.6.4

Multiply $11$ by $8$ .

$264 {(x^{2} + 10)}^{4} x + 12 ((88 x^{2} + 8 \cdot 10) {(x^{2} + 10)}^{3} x)$

Step 2.6.5

Multiply $8$ by $10$ .

$264 {(x^{2} + 10)}^{4} x + 12 ((88 x^{2} + 80) {(x^{2} + 10)}^{3} x)$

Step 2.6.6

Factor $12 {(x^{2} + 10)}^{3} x$ out of $264 {(x^{2} + 10)}^{4} x + 12 (88 x^{2} + 80) {(x^{2} + 10)}^{3} x$ .

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Step 2.6.6.1

Factor $12 {(x^{2} + 10)}^{3} x$ out of $264 {(x^{2} + 10)}^{4} x$ .

$12 {(x^{2} + 10)}^{3} x (22 (x^{2} + 10)) + 12 (88 x^{2} + 80) {(x^{2} + 10)}^{3} x$

Step 2.6.6.2

Factor $12 {(x^{2} + 10)}^{3} x$ out of $12 (88 x^{2} + 80) {(x^{2} + 10)}^{3} x$ .

$12 {(x^{2} + 10)}^{3} x (22 (x^{2} + 10)) + 12 {(x^{2} + 10)}^{3} x (88 x^{2} + 80)$

Step 2.6.6.3

Factor $12 {(x^{2} + 10)}^{3} x$ out of $12 {(x^{2} + 10)}^{3} x (22 (x^{2} + 10)) + 12 {(x^{2} + 10)}^{3} x (88 x^{2} + 80)$ .

$12 {(x^{2} + 10)}^{3} x (22 (x^{2} + 10) + 88 x^{2} + 80)$

Step 2.6.7

Reorder the factors of $12 {(x^{2} + 10)}^{3} x (22 (x^{2} + 10) + 88 x^{2} + 80)$ .

$f'' (x) = 12 x {(x^{2} + 10)}^{3} (22 (x^{2} + 10) + 88 x^{2} + 80)$

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$\frac{d}{d x} [12 x {(x^{2} + 10)}^{3} (22 x^{2} + 22 \cdot 10 + 88 x^{2} + 80)]$

Step 3.1.1.2

Multiply $22$ by $10$ .

$\frac{d}{d x} [12 x {(x^{2} + 10)}^{3} (22 x^{2} + 220 + 88 x^{2} + 80)]$

Step 3.1.2

Simplify by adding terms.

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Step 3.1.2.1

Add $22 x^{2}$ and $88 x^{2}$ .

$\frac{d}{d x} [12 x {(x^{2} + 10)}^{3} (110 x^{2} + 220 + 80)]$

Step 3.1.2.2

Add $220$ and $80$ .

$\frac{d}{d x} [12 x {(x^{2} + 10)}^{3} (110 x^{2} + 300)]$

Step 3.1.3

Since $12$ is constant with respect to $x$ , the derivative of $12 x {(x^{2} + 10)}^{3} (110 x^{2} + 300)$ with respect to $x$ is $12 \frac{d}{d x} [x {(x^{2} + 10)}^{3} (110 x^{2} + 300)]$ .

$12 \frac{d}{d x} [x {(x^{2} + 10)}^{3} (110 x^{2} + 300)]$

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 $f (x) = x {(x^{2} + 10)}^{3}$ and $g (x) = 110 x^{2} + 300$ .

$12 (x {(x^{2} + 10)}^{3} \frac{d}{d x} [110 x^{2} + 300] + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $110 x^{2} + 300$ with respect to $x$ is $\frac{d}{d x} [110 x^{2}] + \frac{d}{d x} [300]$ .

$12 (x {(x^{2} + 10)}^{3} (\frac{d}{d x} [110 x^{2}] + \frac{d}{d x} [300]) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3.2

Since $110$ is constant with respect to $x$ , the derivative of $110 x^{2}$ with respect to $x$ is $110 \frac{d}{d x} [x^{2}]$ .

$12 (x {(x^{2} + 10)}^{3} (110 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [300]) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3.3

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

$12 (x {(x^{2} + 10)}^{3} (110 (2 x) + \frac{d}{d x} [300]) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3.4

Multiply $2$ by $110$ .

$12 (x {(x^{2} + 10)}^{3} (220 x + \frac{d}{d x} [300]) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3.5

Since $300$ is constant with respect to $x$ , the derivative of $300$ with respect to $x$ is $0$ .

$12 (x {(x^{2} + 10)}^{3} (220 x + 0) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.3.6

Add $220 x$ and $0$ .

$12 (x {(x^{2} + 10)}^{3} (220 x) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.4

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

$12 (x^{1} x {(x^{2} + 10)}^{3} \cdot 220 + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.5

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

$12 (x^{1} x^{1} {(x^{2} + 10)}^{3} \cdot 220 + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.6

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

$12 (x^{1 + 1} {(x^{2} + 10)}^{3} \cdot 220 + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $1$ and $1$ .

$12 (x^{2} {(x^{2} + 10)}^{3} \cdot 220 + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.7.2

Move $220$ to the left of $x^{2} {(x^{2} + 10)}^{3}$ .

$12 (220 \cdot (x^{2} {(x^{2} + 10)}^{3}) + (110 x^{2} + 300) \frac{d}{d x} [x {(x^{2} + 10)}^{3}])$

Step 3.8

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x \frac{d}{d x} [{(x^{2} + 10)}^{3}] + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} + 10$ .

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Step 3.9.1

To apply the Chain Rule, set $u$ as $x^{2} + 10$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.9.2

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 u^{2} \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.9.3

Replace all occurrences of $u$ with $x^{2} + 10$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 {(x^{2} + 10)}^{2} \frac{d}{d x} [x^{2} + 10]) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.10

Differentiate.

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Step 3.10.1

By the Sum Rule, the derivative of $x^{2} + 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10]$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 {(x^{2} + 10)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10])) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.10.2

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 {(x^{2} + 10)}^{2} (2 x + \frac{d}{d x} [10])) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.10.3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 {(x^{2} + 10)}^{2} (2 x + 0)) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.10.4

Simplify the expression.

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Step 3.10.4.1

Add $2 x$ and $0$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (3 {(x^{2} + 10)}^{2} (2 x)) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.10.4.2

Multiply $2$ by $3$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x (6 {(x^{2} + 10)}^{2} x) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.11

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{1} x (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.12

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{1} x^{1} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.13

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{1 + 1} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.14

Add $1$ and $1$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3} \frac{d}{d x} [x]))$

Step 3.15

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

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3} \cdot 1))$

Step 3.16

Multiply ${(x^{2} + 10)}^{3}$ by $1$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

$12 (220 x^{2} {(x^{2} + 10)}^{3}) + 12 ((110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.2

Multiply $220$ by $12$ .

$2640 (x^{2} {(x^{2} + 10)}^{3}) + 12 ((110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.3

Factor $12$ out of $2640 x^{2} {(x^{2} + 10)}^{3} + 12 (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3})$ .

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Step 3.17.3.1

Factor $12$ out of $2640 x^{2} {(x^{2} + 10)}^{3}$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3}) + 12 (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3})$

Step 3.17.3.2

Factor $12$ out of $12 (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3})$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3}) + 12 ((110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.3.3

Factor $12$ out of $12 (220 x^{2} {(x^{2} + 10)}^{3}) + 12 ((110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$ .

$12 (220 x^{2} {(x^{2} + 10)}^{3} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4

Simplify each term.

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Step 3.17.4.1

Use the Binomial Theorem.

$12 (220 x^{2} ({(x^{2})}^{3} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2

Simplify each term.

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Step 3.17.4.2.1

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

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Step 3.17.4.2.1.1

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

$12 (220 x^{2} (x^{2 \cdot 3} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.1.2

Multiply $2$ by $3$ .

$12 (220 x^{2} (x^{6} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.2

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

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Step 3.17.4.2.2.1

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

$12 (220 x^{2} (x^{6} + 3 x^{2 \cdot 2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.2.2

Multiply $2$ by $2$ .

$12 (220 x^{2} (x^{6} + 3 x^{4} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.3

Multiply $10$ by $3$ .

$12 (220 x^{2} (x^{6} + 30 x^{4} + 3 x^{2} \cdot 10^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.4

Raise $10$ to the power of $2$ .

$12 (220 x^{2} (x^{6} + 30 x^{4} + 3 x^{2} \cdot 100 + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.5

Multiply $100$ by $3$ .

$12 (220 x^{2} (x^{6} + 30 x^{4} + 300 x^{2} + 10^{3}) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.2.6

Raise $10$ to the power of $3$ .

$12 (220 x^{2} (x^{6} + 30 x^{4} + 300 x^{2} + 1000) + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.3

Apply the distributive property.

$12 (220 x^{2} x^{6} + 220 x^{2} (30 x^{4}) + 220 x^{2} (300 x^{2}) + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4

Simplify.

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Step 3.17.4.4.1

Multiply $x^{2}$ by $x^{6}$ by adding the exponents.

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Step 3.17.4.4.1.1

Move $x^{6}$ .

$12 (220 (x^{6} x^{2}) + 220 x^{2} (30 x^{4}) + 220 x^{2} (300 x^{2}) + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4.1.2

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

$12 (220 x^{6 + 2} + 220 x^{2} (30 x^{4}) + 220 x^{2} (300 x^{2}) + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4.1.3

Add $6$ and $2$ .

$12 (220 x^{8} + 220 x^{2} (30 x^{4}) + 220 x^{2} (300 x^{2}) + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4.2

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 220 \cdot 30 x^{2} x^{4} + 220 x^{2} (300 x^{2}) + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4.3

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 220 \cdot 30 x^{2} x^{4} + 220 \cdot 300 x^{2} x^{2} + 220 x^{2} \cdot 1000 + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.4.4

Multiply $1000$ by $220$ .

$12 (220 x^{8} + 220 \cdot 30 x^{2} x^{4} + 220 \cdot 300 x^{2} x^{2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5

Simplify each term.

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Step 3.17.4.5.1

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.17.4.5.1.1

Move $x^{4}$ .

$12 (220 x^{8} + 220 \cdot 30 (x^{4} x^{2}) + 220 \cdot 300 x^{2} x^{2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.1.2

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

$12 (220 x^{8} + 220 \cdot 30 x^{4 + 2} + 220 \cdot 300 x^{2} x^{2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.1.3

Add $4$ and $2$ .

$12 (220 x^{8} + 220 \cdot 30 x^{6} + 220 \cdot 300 x^{2} x^{2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.2

Multiply $220$ by $30$ .

$12 (220 x^{8} + 6600 x^{6} + 220 \cdot 300 x^{2} x^{2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.3

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

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Step 3.17.4.5.3.1

Move $x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 220 \cdot 300 (x^{2} x^{2}) + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.3.2

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

$12 (220 x^{8} + 6600 x^{6} + 220 \cdot 300 x^{2 + 2} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.3.3

Add $2$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 220 \cdot 300 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.5.4

Multiply $220$ by $300$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (x^{2} (6 {(x^{2} + 10)}^{2}) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6

Simplify each term.

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Step 3.17.4.6.1

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} {(x^{2} + 10)}^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.2

Rewrite ${(x^{2} + 10)}^{2}$ as $(x^{2} + 10) (x^{2} + 10)$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} ((x^{2} + 10) (x^{2} + 10)) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.3

Expand $(x^{2} + 10) (x^{2} + 10)$ using the FOIL Method.

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Step 3.17.4.6.3.1

Apply the distributive property.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{2} (x^{2} + 10) + 10 (x^{2} + 10)) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.3.2

Apply the distributive property.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{2} x^{2} + x^{2} \cdot 10 + 10 (x^{2} + 10)) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.3.3

Apply the distributive property.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{2} x^{2} + x^{2} \cdot 10 + 10 x^{2} + 10 \cdot 10) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.4

Simplify and combine like terms.

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Step 3.17.4.6.4.1

Simplify each term.

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Step 3.17.4.6.4.1.1

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

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Step 3.17.4.6.4.1.1.1

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{2 + 2} + x^{2} \cdot 10 + 10 x^{2} + 10 \cdot 10) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.4.1.1.2

Add $2$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{4} + x^{2} \cdot 10 + 10 x^{2} + 10 \cdot 10) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.4.1.2

Move $10$ to the left of $x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{4} + 10 \cdot x^{2} + 10 x^{2} + 10 \cdot 10) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.4.1.3

Multiply $10$ by $10$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{4} + 10 x^{2} + 10 x^{2} + 100) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.4.2

Add $10 x^{2}$ and $10 x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} (x^{4} + 20 x^{2} + 100) + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.5

Apply the distributive property.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{2} x^{4} + 6 x^{2} (20 x^{2}) + 6 x^{2} \cdot 100 + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.6

Simplify.

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Step 3.17.4.6.6.1

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.17.4.6.6.1.1

Move $x^{4}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 (x^{4} x^{2}) + 6 x^{2} (20 x^{2}) + 6 x^{2} \cdot 100 + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.6.1.2

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{4 + 2} + 6 x^{2} (20 x^{2}) + 6 x^{2} \cdot 100 + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.6.1.3

Add $4$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 x^{2} (20 x^{2}) + 6 x^{2} \cdot 100 + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.6.2

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 \cdot 20 x^{2} x^{2} + 6 x^{2} \cdot 100 + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.6.3

Multiply $100$ by $6$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 \cdot 20 x^{2} x^{2} + 600 x^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.7

Simplify each term.

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Step 3.17.4.6.7.1

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

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Step 3.17.4.6.7.1.1

Move $x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 \cdot 20 (x^{2} x^{2}) + 600 x^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.7.1.2

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 \cdot 20 x^{2 + 2} + 600 x^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.7.1.3

Add $2$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 6 \cdot 20 x^{4} + 600 x^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.7.2

Multiply $6$ by $20$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + {(x^{2} + 10)}^{3}))$

Step 3.17.4.6.8

Use the Binomial Theorem.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + {(x^{2})}^{3} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9

Simplify each term.

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Step 3.17.4.6.9.1

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

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Step 3.17.4.6.9.1.1

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{2 \cdot 3} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9.1.2

Multiply $2$ by $3$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 3 {(x^{2})}^{2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9.2

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

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Step 3.17.4.6.9.2.1

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 3 x^{2 \cdot 2} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9.2.2

Multiply $2$ by $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 3 x^{4} \cdot 10 + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9.3

Multiply $10$ by $3$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 30 x^{4} + 3 x^{2} \cdot 10^{2} + 10^{3}))$

Step 3.17.4.6.9.4

Raise $10$ to the power of $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 30 x^{4} + 3 x^{2} \cdot 100 + 10^{3}))$

Step 3.17.4.6.9.5

Multiply $100$ by $3$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 30 x^{4} + 300 x^{2} + 10^{3}))$

Step 3.17.4.6.9.6

Raise $10$ to the power of $3$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (6 x^{6} + 120 x^{4} + 600 x^{2} + x^{6} + 30 x^{4} + 300 x^{2} + 1000))$

Step 3.17.4.7

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (7 x^{6} + 120 x^{4} + 600 x^{2} + 30 x^{4} + 300 x^{2} + 1000))$

Step 3.17.4.8

Add $120 x^{4}$ and $30 x^{4}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (7 x^{6} + 150 x^{4} + 600 x^{2} + 300 x^{2} + 1000))$

Step 3.17.4.9

Add $600 x^{2}$ and $300 x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + (110 x^{2} + 300) (7 x^{6} + 150 x^{4} + 900 x^{2} + 1000))$

Step 3.17.4.10

Expand $(110 x^{2} + 300) (7 x^{6} + 150 x^{4} + 900 x^{2} + 1000)$ by multiplying each term in the first expression by each term in the second expression.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 110 x^{2} (7 x^{6}) + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11

Simplify each term.

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Step 3.17.4.11.1

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 110 \cdot 7 x^{2} x^{6} + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.2

Multiply $x^{2}$ by $x^{6}$ by adding the exponents.

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Step 3.17.4.11.2.1

Move $x^{6}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 110 \cdot 7 (x^{6} x^{2}) + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.2.2

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 110 \cdot 7 x^{6 + 2} + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.2.3

Add $6$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 110 \cdot 7 x^{8} + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.3

Multiply $110$ by $7$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 110 x^{2} (150 x^{4}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.4

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 110 \cdot 150 x^{2} x^{4} + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.5

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.17.4.11.5.1

Move $x^{4}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 110 \cdot 150 (x^{4} x^{2}) + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.5.2

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 110 \cdot 150 x^{4 + 2} + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.5.3

Add $4$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 110 \cdot 150 x^{6} + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.6

Multiply $110$ by $150$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 110 x^{2} (900 x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.7

Rewrite using the commutative property of multiplication.

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 110 \cdot 900 x^{2} x^{2} + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.8

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

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Step 3.17.4.11.8.1

Move $x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 110 \cdot 900 (x^{2} x^{2}) + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.8.2

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

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 110 \cdot 900 x^{2 + 2} + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.8.3

Add $2$ and $2$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 110 \cdot 900 x^{4} + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.9

Multiply $110$ by $900$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110 x^{2} \cdot 1000 + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.10

Multiply $1000$ by $110$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110000 x^{2} + 300 (7 x^{6}) + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.11

Multiply $7$ by $300$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110000 x^{2} + 2100 x^{6} + 300 (150 x^{4}) + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.12

Multiply $150$ by $300$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110000 x^{2} + 2100 x^{6} + 45000 x^{4} + 300 (900 x^{2}) + 300 \cdot 1000)$

Step 3.17.4.11.13

Multiply $900$ by $300$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110000 x^{2} + 2100 x^{6} + 45000 x^{4} + 270000 x^{2} + 300 \cdot 1000)$

Step 3.17.4.11.14

Multiply $300$ by $1000$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 16500 x^{6} + 99000 x^{4} + 110000 x^{2} + 2100 x^{6} + 45000 x^{4} + 270000 x^{2} + 300000)$

Step 3.17.4.12

Add $16500 x^{6}$ and $2100 x^{6}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 18600 x^{6} + 99000 x^{4} + 110000 x^{2} + 45000 x^{4} + 270000 x^{2} + 300000)$

Step 3.17.4.13

Add $99000 x^{4}$ and $45000 x^{4}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 18600 x^{6} + 144000 x^{4} + 110000 x^{2} + 270000 x^{2} + 300000)$

Step 3.17.4.14

Add $110000 x^{2}$ and $270000 x^{2}$ .

$12 (220 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 770 x^{8} + 18600 x^{6} + 144000 x^{4} + 380000 x^{2} + 300000)$

Step 3.17.5

Add $220 x^{8}$ and $770 x^{8}$ .

$12 (990 x^{8} + 6600 x^{6} + 66000 x^{4} + 220000 x^{2} + 18600 x^{6} + 144000 x^{4} + 380000 x^{2} + 300000)$

Step 3.17.6

Add $6600 x^{6}$ and $18600 x^{6}$ .

$12 (990 x^{8} + 25200 x^{6} + 66000 x^{4} + 220000 x^{2} + 144000 x^{4} + 380000 x^{2} + 300000)$

Step 3.17.7

Add $66000 x^{4}$ and $144000 x^{4}$ .

$12 (990 x^{8} + 25200 x^{6} + 210000 x^{4} + 220000 x^{2} + 380000 x^{2} + 300000)$

Step 3.17.8

Add $220000 x^{2}$ and $380000 x^{2}$ .

$12 (990 x^{8} + 25200 x^{6} + 210000 x^{4} + 600000 x^{2} + 300000)$

Step 3.17.9

Apply the distributive property.

$12 (990 x^{8}) + 12 (25200 x^{6}) + 12 (210000 x^{4}) + 12 (600000 x^{2}) + 12 \cdot 300000$

Step 3.17.10

Simplify.

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Step 3.17.10.1

Multiply $990$ by $12$ .

$11880 x^{8} + 12 (25200 x^{6}) + 12 (210000 x^{4}) + 12 (600000 x^{2}) + 12 \cdot 300000$

Step 3.17.10.2

Multiply $25200$ by $12$ .

$11880 x^{8} + 302400 x^{6} + 12 (210000 x^{4}) + 12 (600000 x^{2}) + 12 \cdot 300000$

Step 3.17.10.3

Multiply $210000$ by $12$ .

$11880 x^{8} + 302400 x^{6} + 2520000 x^{4} + 12 (600000 x^{2}) + 12 \cdot 300000$

Step 3.17.10.4

Multiply $600000$ by $12$ .

$11880 x^{8} + 302400 x^{6} + 2520000 x^{4} + 7200000 x^{2} + 12 \cdot 300000$

Step 3.17.10.5

Multiply $12$ by $300000$ .

$f''' (x) = 11880 x^{8} + 302400 x^{6} + 2520000 x^{4} + 7200000 x^{2} + 3600000$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $11880 x^{8} + 302400 x^{6} + 2520000 x^{4} + 7200000 x^{2} + 3600000$ with respect to $x$ is $\frac{d}{d x} [11880 x^{8}] + \frac{d}{d x} [302400 x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$ .

$\frac{d}{d x} [11880 x^{8}] + \frac{d}{d x} [302400 x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.2

Evaluate $\frac{d}{d x} [11880 x^{8}]$ .

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Step 4.2.1

Since $11880$ is constant with respect to $x$ , the derivative of $11880 x^{8}$ with respect to $x$ is $11880 \frac{d}{d x} [x^{8}]$ .

$11880 \frac{d}{d x} [x^{8}] + \frac{d}{d x} [302400 x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.2.2

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

$11880 (8 x^{7}) + \frac{d}{d x} [302400 x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.2.3

Multiply $8$ by $11880$ .

$95040 x^{7} + \frac{d}{d x} [302400 x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.3

Evaluate $\frac{d}{d x} [302400 x^{6}]$ .

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Step 4.3.1

Since $302400$ is constant with respect to $x$ , the derivative of $302400 x^{6}$ with respect to $x$ is $302400 \frac{d}{d x} [x^{6}]$ .

$95040 x^{7} + 302400 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.3.2

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

$95040 x^{7} + 302400 (6 x^{5}) + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.3.3

Multiply $6$ by $302400$ .

$95040 x^{7} + 1814400 x^{5} + \frac{d}{d x} [2520000 x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.4

Evaluate $\frac{d}{d x} [2520000 x^{4}]$ .

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Step 4.4.1

Since $2520000$ is constant with respect to $x$ , the derivative of $2520000 x^{4}$ with respect to $x$ is $2520000 \frac{d}{d x} [x^{4}]$ .

$95040 x^{7} + 1814400 x^{5} + 2520000 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.4.2

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

$95040 x^{7} + 1814400 x^{5} + 2520000 (4 x^{3}) + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.4.3

Multiply $4$ by $2520000$ .

$95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + \frac{d}{d x} [7200000 x^{2}] + \frac{d}{d x} [3600000]$

Step 4.5

Evaluate $\frac{d}{d x} [7200000 x^{2}]$ .

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Step 4.5.1

Since $7200000$ is constant with respect to $x$ , the derivative of $7200000 x^{2}$ with respect to $x$ is $7200000 \frac{d}{d x} [x^{2}]$ .

$95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 7200000 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3600000]$

Step 4.5.2

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

$95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 7200000 (2 x) + \frac{d}{d x} [3600000]$

Step 4.5.3

Multiply $2$ by $7200000$ .

$95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 14400000 x + \frac{d}{d x} [3600000]$

Step 4.6

Differentiate using the Constant Rule.

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Step 4.6.1

Since $3600000$ is constant with respect to $x$ , the derivative of $3600000$ with respect to $x$ is $0$ .

$95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 14400000 x + 0$

Step 4.6.2

Add $95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 14400000 x$ and $0$ .

$f^{4} (x) = 95040 x^{7} + 1814400 x^{5} + 10080000 x^{3} + 14400000 x$