Calculus Examples

$8 x {(x^{2} + 9)}^{3}$

Step 1

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

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

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 $f (x) = x$ and $g (x) = {(x^{2} + 9)}^{3}$ .

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

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 $g (x) = x^{2} + 9$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} + 9$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x^{2} + 9$ .

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

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

Step 4.2

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

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

Step 4.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $2 x$ and $0$ .

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

Step 4.4.2

Multiply $2$ by $3$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

$8 (x^{2} (6 {(x^{2} + 9)}^{2}) + {(x^{2} + 9)}^{3} \cdot 1)$

Step 10

Multiply ${(x^{2} + 9)}^{3}$ by $1$ .

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Multiply $6$ by $8$ .

$48 (x^{2} ({(x^{2} + 9)}^{2})) + 8 {(x^{2} + 9)}^{3}$

Step 11.3

Factor $8 {(x^{2} + 9)}^{2}$ out of $48 x^{2} {(x^{2} + 9)}^{2} + 8 {(x^{2} + 9)}^{3}$ .

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Step 11.3.1

Factor $8 {(x^{2} + 9)}^{2}$ out of $48 x^{2} {(x^{2} + 9)}^{2}$ .

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

Step 11.3.2

Factor $8 {(x^{2} + 9)}^{2}$ out of $8 {(x^{2} + 9)}^{3}$ .

$8 {(x^{2} + 9)}^{2} (6 x^{2}) + 8 {(x^{2} + 9)}^{2} (x^{2} + 9)$

Step 11.3.3

Factor $8 {(x^{2} + 9)}^{2}$ out of $8 {(x^{2} + 9)}^{2} (6 x^{2}) + 8 {(x^{2} + 9)}^{2} (x^{2} + 9)$ .

$8 {(x^{2} + 9)}^{2} (6 x^{2} + x^{2} + 9)$

Step 11.4

Add $6 x^{2}$ and $x^{2}$ .

$8 {(x^{2} + 9)}^{2} (7 x^{2} + 9)$

Step 11.5

Rewrite ${(x^{2} + 9)}^{2}$ as $(x^{2} + 9) (x^{2} + 9)$ .

$8 ((x^{2} + 9) (x^{2} + 9)) (7 x^{2} + 9)$

Step 11.6

Expand $(x^{2} + 9) (x^{2} + 9)$ using the FOIL Method.

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Step 11.6.1

Apply the distributive property.

$8 (x^{2} (x^{2} + 9) + 9 (x^{2} + 9)) (7 x^{2} + 9)$

Step 11.6.2

Apply the distributive property.

$8 (x^{2} x^{2} + x^{2} \cdot 9 + 9 (x^{2} + 9)) (7 x^{2} + 9)$

Step 11.6.3

Apply the distributive property.

$8 (x^{2} x^{2} + x^{2} \cdot 9 + 9 x^{2} + 9 \cdot 9) (7 x^{2} + 9)$

Step 11.7

Simplify and combine like terms.

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Step 11.7.1

Simplify each term.

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Step 11.7.1.1

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

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Step 11.7.1.1.1

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

$8 (x^{2 + 2} + x^{2} \cdot 9 + 9 x^{2} + 9 \cdot 9) (7 x^{2} + 9)$

Step 11.7.1.1.2

Add $2$ and $2$ .

$8 (x^{4} + x^{2} \cdot 9 + 9 x^{2} + 9 \cdot 9) (7 x^{2} + 9)$

Step 11.7.1.2

Move $9$ to the left of $x^{2}$ .

$8 (x^{4} + 9 \cdot x^{2} + 9 x^{2} + 9 \cdot 9) (7 x^{2} + 9)$

Step 11.7.1.3

Multiply $9$ by $9$ .

$8 (x^{4} + 9 x^{2} + 9 x^{2} + 81) (7 x^{2} + 9)$

Step 11.7.2

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

$8 (x^{4} + 18 x^{2} + 81) (7 x^{2} + 9)$

Step 11.8

Apply the distributive property.

$(8 x^{4} + 8 (18 x^{2}) + 8 \cdot 81) (7 x^{2} + 9)$

Step 11.9

Simplify.

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Step 11.9.1

Multiply $18$ by $8$ .

$(8 x^{4} + 144 x^{2} + 8 \cdot 81) (7 x^{2} + 9)$

Step 11.9.2

Multiply $8$ by $81$ .

$(8 x^{4} + 144 x^{2} + 648) (7 x^{2} + 9)$

Step 11.10

Expand $(8 x^{4} + 144 x^{2} + 648) (7 x^{2} + 9)$ by multiplying each term in the first expression by each term in the second expression.

$8 x^{4} (7 x^{2}) + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11

Simplify each term.

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Step 11.11.1

Rewrite using the commutative property of multiplication.

$8 \cdot 7 x^{4} x^{2} + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.2

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

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Step 11.11.2.1

Move $x^{2}$ .

$8 \cdot 7 (x^{2} x^{4}) + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.2.2

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

$8 \cdot 7 x^{2 + 4} + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.2.3

Add $2$ and $4$ .

$8 \cdot 7 x^{6} + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.3

Multiply $8$ by $7$ .

$56 x^{6} + 8 x^{4} \cdot 9 + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.4

Multiply $9$ by $8$ .

$56 x^{6} + 72 x^{4} + 144 x^{2} (7 x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.5

Rewrite using the commutative property of multiplication.

$56 x^{6} + 72 x^{4} + 144 \cdot 7 x^{2} x^{2} + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.6

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

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Step 11.11.6.1

Move $x^{2}$ .

$56 x^{6} + 72 x^{4} + 144 \cdot 7 (x^{2} x^{2}) + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.6.2

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

$56 x^{6} + 72 x^{4} + 144 \cdot 7 x^{2 + 2} + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.6.3

Add $2$ and $2$ .

$56 x^{6} + 72 x^{4} + 144 \cdot 7 x^{4} + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.7

Multiply $144$ by $7$ .

$56 x^{6} + 72 x^{4} + 1008 x^{4} + 144 x^{2} \cdot 9 + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.8

Multiply $9$ by $144$ .

$56 x^{6} + 72 x^{4} + 1008 x^{4} + 1296 x^{2} + 648 (7 x^{2}) + 648 \cdot 9$

Step 11.11.9

Multiply $7$ by $648$ .

$56 x^{6} + 72 x^{4} + 1008 x^{4} + 1296 x^{2} + 4536 x^{2} + 648 \cdot 9$

Step 11.11.10

Multiply $648$ by $9$ .

$56 x^{6} + 72 x^{4} + 1008 x^{4} + 1296 x^{2} + 4536 x^{2} + 5832$

Step 11.12

Add $72 x^{4}$ and $1008 x^{4}$ .

$56 x^{6} + 1080 x^{4} + 1296 x^{2} + 4536 x^{2} + 5832$

Step 11.13

Add $1296 x^{2}$ and $4536 x^{2}$ .

$56 x^{6} + 1080 x^{4} + 5832 x^{2} + 5832$