Calculus Examples

$f (x) = {(3 x^{2} - 8 x + 80)}^{9}$

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 $g (x) = 3 x^{2} - 8 x + 80$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 8 x + 80$ .

$\frac{d}{d u} [u^{9}] \frac{d}{d x} [3 x^{2} - 8 x + 80]$

Step 1.1.2

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

$9 u^{8} \frac{d}{d x} [3 x^{2} - 8 x + 80]$

Step 1.1.3

Replace all occurrences of $u$ with $3 x^{2} - 8 x + 80$ .

$9 {(3 x^{2} - 8 x + 80)}^{8} \frac{d}{d x} [3 x^{2} - 8 x + 80]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x^{2} - 8 x + 80$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]$ .

$9 {(3 x^{2} - 8 x + 80)}^{8} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80])$

Step 1.2.2

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

$9 {(3 x^{2} - 8 x + 80)}^{8} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80])$

Step 1.2.3

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

$9 {(3 x^{2} - 8 x + 80)}^{8} (3 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80])$

Step 1.2.4

Multiply $2$ by $3$ .

$9 {(3 x^{2} - 8 x + 80)}^{8} (6 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80])$

Step 1.2.5

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

$9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [80])$

Step 1.2.6

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

$9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8 \cdot 1 + \frac{d}{d x} [80])$

Step 1.2.7

Multiply $- 8$ by $1$ .

$9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8 + \frac{d}{d x} [80])$

Step 1.2.8

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8 + 0)$

Step 1.2.9

Add $6 x - 8$ and $0$ .

$f' (x) = 9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8)$

Step 2

Find the second derivative.

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Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 {(3 x^{2} - 8 x + 80)}^{8} (6 x - 8)$ with respect to $x$ is $9 \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8} (6 x - 8)]$ .

$9 \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8} (6 x - 8)]$

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 $f (x) = {(3 x^{2} - 8 x + 80)}^{8}$ and $g (x) = 6 x - 8$ .

$9 ({(3 x^{2} - 8 x + 80)}^{8} \frac{d}{d x} [6 x - 8] + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $6 x - 8$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 8]$ .

$9 ({(3 x^{2} - 8 x + 80)}^{8} (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 8]) + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.2

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

$9 ({(3 x^{2} - 8 x + 80)}^{8} (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.3

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

$9 ({(3 x^{2} - 8 x + 80)}^{8} (6 \cdot 1 + \frac{d}{d x} [- 8]) + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.4

Multiply $6$ by $1$ .

$9 ({(3 x^{2} - 8 x + 80)}^{8} (6 + \frac{d}{d x} [- 8]) + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.5

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

$9 ({(3 x^{2} - 8 x + 80)}^{8} (6 + 0) + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $6$ and $0$ .

$9 ({(3 x^{2} - 8 x + 80)}^{8} \cdot 6 + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

Step 2.3.6.2

Move $6$ to the left of ${(3 x^{2} - 8 x + 80)}^{8}$ .

$9 (6 \cdot {(3 x^{2} - 8 x + 80)}^{8} + (6 x - 8) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{8}])$

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 $f (x) = x^{8}$ and $g (x) = 3 x^{2} - 8 x + 80$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 8 x + 80$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + (6 x - 8) (\frac{d}{d u} [u^{8}] \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 2.4.2

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + (6 x - 8) (8 u^{7} \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 2.4.3

Replace all occurrences of $u$ with $3 x^{2} - 8 x + 80$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + (6 x - 8) (8 {(3 x^{2} - 8 x + 80)}^{7} \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 2.5

Differentiate.

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Step 2.5.1

Move $8$ to the left of $6 x - 8$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 \cdot (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} \frac{d}{d x} [3 x^{2} - 8 x + 80])$

Step 2.5.2

By the Sum Rule, the derivative of $3 x^{2} - 8 x + 80$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 2.5.3

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 2.5.4

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (3 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 2.5.5

Multiply $2$ by $3$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 2.5.6

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [80]))$

Step 2.5.7

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x - 8 \cdot 1 + \frac{d}{d x} [80]))$

Step 2.5.8

Multiply $- 8$ by $1$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x - 8 + \frac{d}{d x} [80]))$

Step 2.5.9

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x - 8 + 0))$

Step 2.5.10

Add $6 x - 8$ and $0$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 (6 x - 8) {(3 x^{2} - 8 x + 80)}^{7} (6 x - 8))$

Step 2.6

Raise $6 x - 8$ to the power of $1$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 ({(6 x - 8)}^{1} (6 x - 8)) {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.7

Raise $6 x - 8$ to the power of $1$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 ({(6 x - 8)}^{1} {(6 x - 8)}^{1}) {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.8

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

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 {(6 x - 8)}^{1 + 1} {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.9

Add $1$ and $1$ .

$9 (6 {(3 x^{2} - 8 x + 80)}^{8} + 8 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$9 (6 {(3 x^{2} - 8 x + 80)}^{8}) + 9 (8 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.10.2

Multiply $6$ by $9$ .

$54 {(3 x^{2} - 8 x + 80)}^{8} + 9 (8 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.10.3

Multiply $8$ by $9$ .

$54 {(3 x^{2} - 8 x + 80)}^{8} + 72 ({(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7})$

Step 2.10.4

Factor $18 {(3 x^{2} - 8 x + 80)}^{7}$ out of $54 {(3 x^{2} - 8 x + 80)}^{8} + 72 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7}$ .

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Step 2.10.4.1

Factor $18 {(3 x^{2} - 8 x + 80)}^{7}$ out of $54 {(3 x^{2} - 8 x + 80)}^{8}$ .

$18 {(3 x^{2} - 8 x + 80)}^{7} (3 (3 x^{2} - 8 x + 80)) + 72 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7}$

Step 2.10.4.2

Factor $18 {(3 x^{2} - 8 x + 80)}^{7}$ out of $72 {(6 x - 8)}^{2} {(3 x^{2} - 8 x + 80)}^{7}$ .

$18 {(3 x^{2} - 8 x + 80)}^{7} (3 (3 x^{2} - 8 x + 80)) + 18 {(3 x^{2} - 8 x + 80)}^{7} (4 {(6 x - 8)}^{2})$

Step 2.10.4.3

Factor $18 {(3 x^{2} - 8 x + 80)}^{7}$ out of $18 {(3 x^{2} - 8 x + 80)}^{7} (3 (3 x^{2} - 8 x + 80)) + 18 {(3 x^{2} - 8 x + 80)}^{7} (4 {(6 x - 8)}^{2})$ .

$f'' (x) = 18 {(3 x^{2} - 8 x + 80)}^{7} (3 (3 x^{2} - 8 x + 80) + 4 {(6 x - 8)}^{2})$

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.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (3 (3 x^{2}) + 3 (- 8 x) + 3 \cdot 80 + 4 {(6 x - 8)}^{2})]$

Step 3.1.1.2

Simplify.

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Step 3.1.1.2.1

Multiply $3$ by $3$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} + 3 (- 8 x) + 3 \cdot 80 + 4 {(6 x - 8)}^{2})]$

Step 3.1.1.2.2

Multiply $- 8$ by $3$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 3 \cdot 80 + 4 {(6 x - 8)}^{2})]$

Step 3.1.1.2.3

Multiply $3$ by $80$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 {(6 x - 8)}^{2})]$

Step 3.1.1.3

Rewrite ${(6 x - 8)}^{2}$ as $(6 x - 8) (6 x - 8)$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 ((6 x - 8) (6 x - 8)))]$

Step 3.1.1.4

Expand $(6 x - 8) (6 x - 8)$ using the FOIL Method.

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Step 3.1.1.4.1

Apply the distributive property.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 x (6 x - 8) - 8 (6 x - 8)))]$

Step 3.1.1.4.2

Apply the distributive property.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 x (6 x) + 6 x \cdot - 8 - 8 (6 x - 8)))]$

Step 3.1.1.4.3

Apply the distributive property.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 x (6 x) + 6 x \cdot - 8 - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5

Simplify and combine like terms.

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Step 3.1.1.5.1

Simplify each term.

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Step 3.1.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 \cdot 6 x \cdot x + 6 x \cdot - 8 - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.1.5.1.2.1

Move $x$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 \cdot 6 (x \cdot x) + 6 x \cdot - 8 - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (6 \cdot 6 x^{2} + 6 x \cdot - 8 - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5.1.3

Multiply $6$ by $6$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2} + 6 x \cdot - 8 - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5.1.4

Multiply $- 8$ by $6$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2} - 48 x - 8 (6 x) - 8 \cdot - 8))]$

Step 3.1.1.5.1.5

Multiply $6$ by $- 8$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2} - 48 x - 48 x - 8 \cdot - 8))]$

Step 3.1.1.5.1.6

Multiply $- 8$ by $- 8$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2} - 48 x - 48 x + 64))]$

Step 3.1.1.5.2

Subtract $48 x$ from $- 48 x$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2} - 96 x + 64))]$

Step 3.1.1.6

Apply the distributive property.

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 4 (36 x^{2}) + 4 (- 96 x) + 4 \cdot 64)]$

Step 3.1.1.7

Simplify.

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Step 3.1.1.7.1

Multiply $36$ by $4$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 144 x^{2} + 4 (- 96 x) + 4 \cdot 64)]$

Step 3.1.1.7.2

Multiply $- 96$ by $4$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 144 x^{2} - 384 x + 4 \cdot 64)]$

Step 3.1.1.7.3

Multiply $4$ by $64$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (9 x^{2} - 24 x + 240 + 144 x^{2} - 384 x + 256)]$

Step 3.1.2

Simplify by adding terms.

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Step 3.1.2.1

Add $9 x^{2}$ and $144 x^{2}$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 24 x + 240 - 384 x + 256)]$

Step 3.1.2.2

Subtract $384 x$ from $- 24 x$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 408 x + 240 + 256)]$

Step 3.1.2.3

Add $240$ and $256$ .

$\frac{d}{d x} [18 {(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 408 x + 496)]$

Step 3.1.3

Since $18$ is constant with respect to $x$ , the derivative of $18 {(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 408 x + 496)$ with respect to $x$ is $18 \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 408 x + 496)]$ .

$18 \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7} (153 x^{2} - 408 x + 496)]$

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 $f (x) = {(3 x^{2} - 8 x + 80)}^{7}$ and $g (x) = 153 x^{2} - 408 x + 496$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} \frac{d}{d x} [153 x^{2} - 408 x + 496] + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $153 x^{2} - 408 x + 496$ with respect to $x$ is $\frac{d}{d x} [153 x^{2}] + \frac{d}{d x} [- 408 x] + \frac{d}{d x} [496]$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (\frac{d}{d x} [153 x^{2}] + \frac{d}{d x} [- 408 x] + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.2

Since $153$ is constant with respect to $x$ , the derivative of $153 x^{2}$ with respect to $x$ is $153 \frac{d}{d x} [x^{2}]$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (153 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 408 x] + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.3

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (153 (2 x) + \frac{d}{d x} [- 408 x] + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.4

Multiply $2$ by $153$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x + \frac{d}{d x} [- 408 x] + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.5

Since $- 408$ is constant with respect to $x$ , the derivative of $- 408 x$ with respect to $x$ is $- 408 \frac{d}{d x} [x]$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408 \frac{d}{d x} [x] + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.6

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408 \cdot 1 + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.7

Multiply $- 408$ by $1$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408 + \frac{d}{d x} [496]) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.8

Since $496$ is constant with respect to $x$ , the derivative of $496$ with respect to $x$ is $0$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408 + 0) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

Step 3.3.9

Add $306 x - 408$ and $0$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + (153 x^{2} - 408 x + 496) \frac{d}{d x} [{(3 x^{2} - 8 x + 80)}^{7}])$

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 $g (x) = 3 x^{2} - 8 x + 80$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 8 x + 80$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + (153 x^{2} - 408 x + 496) (\frac{d}{d u} [u^{7}] \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 3.4.2

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + (153 x^{2} - 408 x + 496) (7 u^{6} \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 3.4.3

Replace all occurrences of $u$ with $3 x^{2} - 8 x + 80$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + (153 x^{2} - 408 x + 496) (7 {(3 x^{2} - 8 x + 80)}^{6} \frac{d}{d x} [3 x^{2} - 8 x + 80]))$

Step 3.5

Differentiate.

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Step 3.5.1

Move $7$ to the left of $153 x^{2} - 408 x + 496$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 \cdot (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} \frac{d}{d x} [3 x^{2} - 8 x + 80])$

Step 3.5.2

By the Sum Rule, the derivative of $3 x^{2} - 8 x + 80$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 3.5.3

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 3.5.4

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (3 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 3.5.5

Multiply $2$ by $3$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [80]))$

Step 3.5.6

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [80]))$

Step 3.5.7

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

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8 \cdot 1 + \frac{d}{d x} [80]))$

Step 3.5.8

Multiply $- 8$ by $1$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8 + \frac{d}{d x} [80]))$

Step 3.5.9

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8 + 0))$

Step 3.5.10

Add $6 x - 8$ and $0$ .

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 7 (153 x^{2} - 408 x + 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + (7 (153 x^{2}) + 7 (- 408 x) + 7 \cdot 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6.2

Apply the distributive property.

$18 ({(3 x^{2} - 8 x + 80)}^{7} (306 x - 408)) + 18 ((7 (153 x^{2}) + 7 (- 408 x) + 7 \cdot 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6.3

Multiply $153$ by $7$ .

$18 {(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 18 ((1071 x^{2} + 7 (- 408 x) + 7 \cdot 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6.4

Multiply $- 408$ by $7$ .

$18 {(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 18 ((1071 x^{2} - 2856 x + 7 \cdot 496) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6.5

Multiply $7$ by $496$ .

$18 {(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 18 ((1071 x^{2} - 2856 x + 3472) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8))$

Step 3.6.6

Factor $18 {(3 x^{2} - 8 x + 80)}^{6}$ out of $18 {(3 x^{2} - 8 x + 80)}^{7} (306 x - 408) + 18 (1071 x^{2} - 2856 x + 3472) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8)$ .

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Step 3.6.6.1

Factor $18 {(3 x^{2} - 8 x + 80)}^{6}$ out of $18 {(3 x^{2} - 8 x + 80)}^{7} (306 x - 408)$ .

$18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408)) + 18 (1071 x^{2} - 2856 x + 3472) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8)$

Step 3.6.6.2

Factor $18 {(3 x^{2} - 8 x + 80)}^{6}$ out of $18 (1071 x^{2} - 2856 x + 3472) {(3 x^{2} - 8 x + 80)}^{6} (6 x - 8)$ .

$18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408)) + 18 {(3 x^{2} - 8 x + 80)}^{6} ((1071 x^{2} - 2856 x + 3472) (6 x - 8))$

Step 3.6.6.3

Factor $18 {(3 x^{2} - 8 x + 80)}^{6}$ out of $18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408)) + 18 {(3 x^{2} - 8 x + 80)}^{6} ((1071 x^{2} - 2856 x + 3472) (6 x - 8))$ .

$f''' (x) = 18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408) + (1071 x^{2} - 2856 x + 3472) (6 x - 8))$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408) + (1071 x^{2} - 2856 x + 3472) (6 x - 8))$ .

$18 {(3 x^{2} - 8 x + 80)}^{6} ((3 x^{2} - 8 x + 80) (306 x - 408) + (1071 x^{2} - 2856 x + 3472) (6 x - 8))$