Calculus Examples

$y = \frac{2}{3} \cdot (x^{3} {(4 x^{5} - 13)}^{3})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $x^{3}$ and $\frac{2}{3}$ .

$\frac{d}{d x} [\frac{x^{3} \cdot 2}{3} \cdot {(4 x^{5} - 13)}^{3}]$

Step 1.2

Combine fractions.

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Step 1.2.1

Combine $\frac{x^{3} \cdot 2}{3}$ and ${(4 x^{5} - 13)}^{3}$ .

$\frac{d}{d x} [\frac{x^{3} \cdot 2 {(4 x^{5} - 13)}^{3}}{3}]$

Step 1.2.2

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

$\frac{d}{d x} [\frac{2 \cdot x^{3} {(4 x^{5} - 13)}^{3}}{3}]$

Step 1.3

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x^{3} {(4 x^{5} - 13)}^{3}}{3}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{3} {(4 x^{5} - 13)}^{3}]$ .

$\frac{2}{3} \frac{d}{d x} [x^{3} {(4 x^{5} - 13)}^{3}]$

Step 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 $f (x) = x^{3}$ and $g (x) = {(4 x^{5} - 13)}^{3}$ .

$\frac{2}{3} (x^{3} \frac{d}{d x} [{(4 x^{5} - 13)}^{3}] + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 4 x^{5} - 13$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 x^{5} - 13$ .

$\frac{2}{3} (x^{3} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 x^{5} - 13]) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 3.2

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

$\frac{2}{3} (x^{3} (3 u^{2} \frac{d}{d x} [4 x^{5} - 13]) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 3.3

Replace all occurrences of $u$ with $4 x^{5} - 13$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} \frac{d}{d x} [4 x^{5} - 13]) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $4 x^{5} - 13$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 13]$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 13])) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{5}$ with respect to $x$ is $4 \frac{d}{d x} [x^{5}]$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (4 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 13])) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.3

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

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (4 (5 x^{4}) + \frac{d}{d x} [- 13])) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.4

Multiply $5$ by $4$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (20 x^{4} + \frac{d}{d x} [- 13])) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.5

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13$ with respect to $x$ is $0$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (20 x^{4} + 0)) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $20 x^{4}$ and $0$ .

$\frac{2}{3} (x^{3} (3 {(4 x^{5} - 13)}^{2} (20 x^{4})) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.6.2

Multiply $20$ by $3$ .

$\frac{2}{3} (x^{3} (60 {(4 x^{5} - 13)}^{2} x^{4}) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 5

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 5.1

Move $x^{4}$ .

$\frac{2}{3} (x^{4} x^{3} (60 {(4 x^{5} - 13)}^{2}) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 5.2

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

$\frac{2}{3} (x^{4 + 3} (60 {(4 x^{5} - 13)}^{2}) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 5.3

Add $4$ and $3$ .

$\frac{2}{3} (x^{7} (60 {(4 x^{5} - 13)}^{2}) + {(4 x^{5} - 13)}^{3} \frac{d}{d x} [x^{3}])$

Step 6

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

$\frac{2}{3} (x^{7} (60 {(4 x^{5} - 13)}^{2}) + {(4 x^{5} - 13)}^{3} (3 x^{2}))$

Step 7

Move $3$ to the left of ${(4 x^{5} - 13)}^{3}$ .

$\frac{2}{3} (x^{7} (60 {(4 x^{5} - 13)}^{2}) + 3 \cdot {(4 x^{5} - 13)}^{3} x^{2})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{2}{3} (x^{7} (60 {(4 x^{5} - 13)}^{2})) + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.2

Combine $x^{7}$ and $\frac{2}{3}$ .

$\frac{x^{7} \cdot 2}{3} (60 {(4 x^{5} - 13)}^{2}) + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.3

Combine $60$ and $\frac{x^{7} \cdot 2}{3}$ .

$\frac{60 (x^{7} \cdot 2)}{3} {(4 x^{5} - 13)}^{2} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.4

Multiply $2$ by $60$ .

$\frac{120 x^{7}}{3} {(4 x^{5} - 13)}^{2} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.5

Combine $\frac{120 x^{7}}{3}$ and ${(4 x^{5} - 13)}^{2}$ .

$\frac{120 x^{7} {(4 x^{5} - 13)}^{2}}{3} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.6

Cancel the common factor of $120$ and $3$ .

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Step 8.6.1

Factor $3$ out of $120 x^{7} {(4 x^{5} - 13)}^{2}$ .

$\frac{3 (40 x^{7} {(4 x^{5} - 13)}^{2})}{3} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.6.2

Cancel the common factors.

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Step 8.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (40 x^{7} {(4 x^{5} - 13)}^{2})}{3 (1)} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.6.2.2

Cancel the common factor.

$\frac{3 (40 x^{7} {(4 x^{5} - 13)}^{2})}{3 \cdot 1} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.6.2.3

Rewrite the expression.

$\frac{40 x^{7} {(4 x^{5} - 13)}^{2}}{1} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.6.2.4

Divide $40 x^{7} {(4 x^{5} - 13)}^{2}$ by $1$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{2}{3} (3 {(4 x^{5} - 13)}^{3} x^{2})$

Step 8.7

Combine $3$ and $\frac{2}{3}$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{3 \cdot 2}{3} ({(4 x^{5} - 13)}^{3} x^{2})$

Step 8.8

Multiply $3$ by $2$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{6}{3} ({(4 x^{5} - 13)}^{3} x^{2})$

Step 8.9

Combine ${(4 x^{5} - 13)}^{3}$ and $\frac{6}{3}$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{{(4 x^{5} - 13)}^{3} \cdot 6}{3} x^{2}$

Step 8.10

Combine $\frac{{(4 x^{5} - 13)}^{3} \cdot 6}{3}$ and $x^{2}$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{{(4 x^{5} - 13)}^{3} \cdot 6 x^{2}}{3}$

Step 8.11

Move $6$ to the left of ${(4 x^{5} - 13)}^{3}$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{6 \cdot {(4 x^{5} - 13)}^{3} x^{2}}{3}$

Step 8.12

Cancel the common factor of $6$ and $3$ .

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Step 8.12.1

Factor $3$ out of $6 {(4 x^{5} - 13)}^{3} x^{2}$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{3 (2 {(4 x^{5} - 13)}^{3} x^{2})}{3}$

Step 8.12.2

Cancel the common factors.

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Step 8.12.2.1

Factor $3$ out of $3$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{3 (2 {(4 x^{5} - 13)}^{3} x^{2})}{3 (1)}$

Step 8.12.2.2

Cancel the common factor.

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{3 (2 {(4 x^{5} - 13)}^{3} x^{2})}{3 \cdot 1}$

Step 8.12.2.3

Rewrite the expression.

$40 x^{7} {(4 x^{5} - 13)}^{2} + \frac{2 {(4 x^{5} - 13)}^{3} x^{2}}{1}$

Step 8.12.2.4

Divide $2 {(4 x^{5} - 13)}^{3} x^{2}$ by $1$ .

$40 x^{7} {(4 x^{5} - 13)}^{2} + 2 {(4 x^{5} - 13)}^{3} x^{2}$

Step 8.13

Factor $2 x^{2} {(4 x^{5} - 13)}^{2}$ out of $40 x^{7} {(4 x^{5} - 13)}^{2} + 2 {(4 x^{5} - 13)}^{3} x^{2}$ .

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Step 8.13.1

Factor $2 x^{2} {(4 x^{5} - 13)}^{2}$ out of $40 x^{7} {(4 x^{5} - 13)}^{2}$ .

$2 x^{2} {(4 x^{5} - 13)}^{2} (20 x^{5}) + 2 {(4 x^{5} - 13)}^{3} x^{2}$

Step 8.13.2

Factor $2 x^{2} {(4 x^{5} - 13)}^{2}$ out of $2 {(4 x^{5} - 13)}^{3} x^{2}$ .

$2 x^{2} {(4 x^{5} - 13)}^{2} (20 x^{5}) + 2 x^{2} {(4 x^{5} - 13)}^{2} (4 x^{5} - 13)$

Step 8.13.3

Factor $2 x^{2} {(4 x^{5} - 13)}^{2}$ out of $2 x^{2} {(4 x^{5} - 13)}^{2} (20 x^{5}) + 2 x^{2} {(4 x^{5} - 13)}^{2} (4 x^{5} - 13)$ .

$2 x^{2} {(4 x^{5} - 13)}^{2} (20 x^{5} + 4 x^{5} - 13)$

Step 8.14

Add $20 x^{5}$ and $4 x^{5}$ .

$2 x^{2} {(4 x^{5} - 13)}^{2} (24 x^{5} - 13)$

Step 8.15

Rewrite ${(4 x^{5} - 13)}^{2}$ as $(4 x^{5} - 13) (4 x^{5} - 13)$ .

$2 x^{2} ((4 x^{5} - 13) (4 x^{5} - 13)) (24 x^{5} - 13)$

Step 8.16

Expand $(4 x^{5} - 13) (4 x^{5} - 13)$ using the FOIL Method.

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Step 8.16.1

Apply the distributive property.

$2 x^{2} (4 x^{5} (4 x^{5} - 13) - 13 (4 x^{5} - 13)) (24 x^{5} - 13)$

Step 8.16.2

Apply the distributive property.

$2 x^{2} (4 x^{5} (4 x^{5}) + 4 x^{5} \cdot - 13 - 13 (4 x^{5} - 13)) (24 x^{5} - 13)$

Step 8.16.3

Apply the distributive property.

$2 x^{2} (4 x^{5} (4 x^{5}) + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17

Simplify and combine like terms.

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Step 8.17.1

Simplify each term.

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Step 8.17.1.1

Rewrite using the commutative property of multiplication.

$2 x^{2} (4 \cdot 4 x^{5} x^{5} + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.2

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 8.17.1.2.1

Move $x^{5}$ .

$2 x^{2} (4 \cdot 4 (x^{5} x^{5}) + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.2.2

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

$2 x^{2} (4 \cdot 4 x^{5 + 5} + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.2.3

Add $5$ and $5$ .

$2 x^{2} (4 \cdot 4 x^{10} + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.3

Multiply $4$ by $4$ .

$2 x^{2} (16 x^{10} + 4 x^{5} \cdot - 13 - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.4

Multiply $- 13$ by $4$ .

$2 x^{2} (16 x^{10} - 52 x^{5} - 13 (4 x^{5}) - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.5

Multiply $4$ by $- 13$ .

$2 x^{2} (16 x^{10} - 52 x^{5} - 52 x^{5} - 13 \cdot - 13) (24 x^{5} - 13)$

Step 8.17.1.6

Multiply $- 13$ by $- 13$ .

$2 x^{2} (16 x^{10} - 52 x^{5} - 52 x^{5} + 169) (24 x^{5} - 13)$

Step 8.17.2

Subtract $52 x^{5}$ from $- 52 x^{5}$ .

$2 x^{2} (16 x^{10} - 104 x^{5} + 169) (24 x^{5} - 13)$

Step 8.18

Apply the distributive property.

$(2 x^{2} (16 x^{10}) + 2 x^{2} (- 104 x^{5}) + 2 x^{2} \cdot 169) (24 x^{5} - 13)$

Step 8.19

Simplify.

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Step 8.19.1

Rewrite using the commutative property of multiplication.

$(2 \cdot 16 x^{2} x^{10} + 2 x^{2} (- 104 x^{5}) + 2 x^{2} \cdot 169) (24 x^{5} - 13)$

Step 8.19.2

Rewrite using the commutative property of multiplication.

$(2 \cdot 16 x^{2} x^{10} + 2 \cdot - 104 x^{2} x^{5} + 2 x^{2} \cdot 169) (24 x^{5} - 13)$

Step 8.19.3

Multiply $169$ by $2$ .

$(2 \cdot 16 x^{2} x^{10} + 2 \cdot - 104 x^{2} x^{5} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20

Simplify each term.

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Step 8.20.1

Multiply $x^{2}$ by $x^{10}$ by adding the exponents.

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Step 8.20.1.1

Move $x^{10}$ .

$(2 \cdot 16 (x^{10} x^{2}) + 2 \cdot - 104 x^{2} x^{5} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.1.2

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

$(2 \cdot 16 x^{10 + 2} + 2 \cdot - 104 x^{2} x^{5} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.1.3

Add $10$ and $2$ .

$(2 \cdot 16 x^{12} + 2 \cdot - 104 x^{2} x^{5} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.2

Multiply $2$ by $16$ .

$(32 x^{12} + 2 \cdot - 104 x^{2} x^{5} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.3

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 8.20.3.1

Move $x^{5}$ .

$(32 x^{12} + 2 \cdot - 104 (x^{5} x^{2}) + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.3.2

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

$(32 x^{12} + 2 \cdot - 104 x^{5 + 2} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.3.3

Add $5$ and $2$ .

$(32 x^{12} + 2 \cdot - 104 x^{7} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.20.4

Multiply $2$ by $- 104$ .

$(32 x^{12} - 208 x^{7} + 338 x^{2}) (24 x^{5} - 13)$

Step 8.21

Expand $(32 x^{12} - 208 x^{7} + 338 x^{2}) (24 x^{5} - 13)$ by multiplying each term in the first expression by each term in the second expression.

$32 x^{12} (24 x^{5}) + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22

Simplify each term.

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Step 8.22.1

Rewrite using the commutative property of multiplication.

$32 \cdot 24 x^{12} x^{5} + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.2

Multiply $x^{12}$ by $x^{5}$ by adding the exponents.

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Step 8.22.2.1

Move $x^{5}$ .

$32 \cdot 24 (x^{5} x^{12}) + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.2.2

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

$32 \cdot 24 x^{5 + 12} + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.2.3

Add $5$ and $12$ .

$32 \cdot 24 x^{17} + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.3

Multiply $32$ by $24$ .

$768 x^{17} + 32 x^{12} \cdot - 13 - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.4

Multiply $- 13$ by $32$ .

$768 x^{17} - 416 x^{12} - 208 x^{7} (24 x^{5}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.5

Rewrite using the commutative property of multiplication.

$768 x^{17} - 416 x^{12} - 208 \cdot 24 x^{7} x^{5} - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.6

Multiply $x^{7}$ by $x^{5}$ by adding the exponents.

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Step 8.22.6.1

Move $x^{5}$ .

$768 x^{17} - 416 x^{12} - 208 \cdot 24 (x^{5} x^{7}) - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.6.2

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

$768 x^{17} - 416 x^{12} - 208 \cdot 24 x^{5 + 7} - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.6.3

Add $5$ and $7$ .

$768 x^{17} - 416 x^{12} - 208 \cdot 24 x^{12} - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.7

Multiply $- 208$ by $24$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} - 208 x^{7} \cdot - 13 + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.8

Multiply $- 13$ by $- 208$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 338 x^{2} (24 x^{5}) + 338 x^{2} \cdot - 13$

Step 8.22.9

Rewrite using the commutative property of multiplication.

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 338 \cdot 24 x^{2} x^{5} + 338 x^{2} \cdot - 13$

Step 8.22.10

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 8.22.10.1

Move $x^{5}$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 338 \cdot 24 (x^{5} x^{2}) + 338 x^{2} \cdot - 13$

Step 8.22.10.2

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

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 338 \cdot 24 x^{5 + 2} + 338 x^{2} \cdot - 13$

Step 8.22.10.3

Add $5$ and $2$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 338 \cdot 24 x^{7} + 338 x^{2} \cdot - 13$

Step 8.22.11

Multiply $338$ by $24$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 8112 x^{7} + 338 x^{2} \cdot - 13$

Step 8.22.12

Multiply $- 13$ by $338$ .

$768 x^{17} - 416 x^{12} - 4992 x^{12} + 2704 x^{7} + 8112 x^{7} - 4394 x^{2}$

Step 8.23

Subtract $4992 x^{12}$ from $- 416 x^{12}$ .

$768 x^{17} - 5408 x^{12} + 2704 x^{7} + 8112 x^{7} - 4394 x^{2}$

Step 8.24

Add $2704 x^{7}$ and $8112 x^{7}$ .

$768 x^{17} - 5408 x^{12} + 10816 x^{7} - 4394 x^{2}$