Calculus Examples

$\sin^{2} (4 x + 9)$

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 $f (x) = x^{2}$ and $g (x) = \sin (4 x + 9)$ .

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Step 1.1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (4 x + 9)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (4 x + 9)]$

Step 1.1.2

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

$2 u_{1} \frac{d}{d x} [\sin (4 x + 9)]$

Step 1.1.3

Replace all occurrences of $u_{1}$ with $\sin (4 x + 9)$ .

$2 \sin (4 x + 9) \frac{d}{d x} [\sin (4 x + 9)]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x + 9$ .

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Step 1.2.1

To apply the Chain Rule, set $u_{2}$ as $4 x + 9$ .

$2 \sin (4 x + 9) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [4 x + 9])$

Step 1.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$2 \sin (4 x + 9) (\cos (u_{2}) \frac{d}{d x} [4 x + 9])$

Step 1.2.3

Replace all occurrences of $u_{2}$ with $4 x + 9$ .

$2 \sin (4 x + 9) (\cos (4 x + 9) \frac{d}{d x} [4 x + 9])$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$2 \sin (4 x + 9) \cos (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9])$

Step 1.3.2

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

$2 \sin (4 x + 9) \cos (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 1.3.3

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

$2 \sin (4 x + 9) \cos (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9])$

Step 1.3.4

Multiply $4$ by $1$ .

$2 \sin (4 x + 9) \cos (4 x + 9) (4 + \frac{d}{d x} [9])$

Step 1.3.5

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

$2 \sin (4 x + 9) \cos (4 x + 9) (4 + 0)$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $4$ and $0$ .

$2 \sin (4 x + 9) \cos (4 x + 9) \cdot 4$

Step 1.3.6.2

Multiply $4$ by $2$ .

$8 \sin (4 x + 9) \cos (4 x + 9)$

Step 1.3.6.3

Reorder the factors of $8 \sin (4 x + 9) \cos (4 x + 9)$ .

$f' (x) = 8 \cos (4 x + 9) \sin (4 x + 9)$

Step 2

Find the second derivative.

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \cos (4 x + 9) \sin (4 x + 9)$ with respect to $x$ is $8 \frac{d}{d x} [\cos (4 x + 9) \sin (4 x + 9)]$ .

$8 \frac{d}{d x} [\cos (4 x + 9) \sin (4 x + 9)]$

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 $f (x) = \cos (4 x + 9)$ and $g (x) = \sin (4 x + 9)$ .

$8 (\cos (4 x + 9) \frac{d}{d x} [\sin (4 x + 9)] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x + 9$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $4 x + 9$ .

$8 (\cos (4 x + 9) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$8 (\cos (4 x + 9) (\cos (u_{1}) \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $4 x + 9$ .

$8 (\cos (4 x + 9) (\cos (4 x + 9) \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.4

Raise $\cos (4 x + 9)$ to the power of $1$ .

$8 (\cos^{1} (4 x + 9) \cos (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.5

Raise $\cos (4 x + 9)$ to the power of $1$ .

$8 (\cos^{1} (4 x + 9) \cos^{1} (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.6

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

$8 ({\cos (4 x + 9)}^{1 + 1} \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7

Differentiate.

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Step 2.7.1

Add $1$ and $1$ .

$8 (\cos^{2} (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.2

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$8 (\cos^{2} (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.3

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

$8 (\cos^{2} (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.4

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

$8 (\cos^{2} (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.5

Multiply $4$ by $1$ .

$8 (\cos^{2} (4 x + 9) (4 + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.6

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

$8 (\cos^{2} (4 x + 9) (4 + 0) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.7

Simplify the expression.

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Step 2.7.7.1

Add $4$ and $0$ .

$8 (\cos^{2} (4 x + 9) \cdot 4 + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.7.7.2

Move $4$ to the left of $\cos^{2} (4 x + 9)$ .

$8 (4 \cdot \cos^{2} (4 x + 9) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 2.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x + 9$ .

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Step 2.8.1

To apply the Chain Rule, set $u_{2}$ as $4 x + 9$ .

$8 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [4 x + 9]))$

Step 2.8.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$8 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (- \sin (u_{2}) \frac{d}{d x} [4 x + 9]))$

Step 2.8.3

Replace all occurrences of $u_{2}$ with $4 x + 9$ .

$8 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (- \sin (4 x + 9) \frac{d}{d x} [4 x + 9]))$

Step 2.9

Raise $\sin (4 x + 9)$ to the power of $1$ .

$8 (4 \cos^{2} (4 x + 9) - (\sin^{1} (4 x + 9) \sin (4 x + 9)) \frac{d}{d x} [4 x + 9])$

Step 2.10

Raise $\sin (4 x + 9)$ to the power of $1$ .

$8 (4 \cos^{2} (4 x + 9) - (\sin^{1} (4 x + 9) \sin^{1} (4 x + 9)) \frac{d}{d x} [4 x + 9])$

Step 2.11

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

$8 (4 \cos^{2} (4 x + 9) - {\sin (4 x + 9)}^{1 + 1} \frac{d}{d x} [4 x + 9])$

Step 2.12

Add $1$ and $1$ .

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) \frac{d}{d x} [4 x + 9])$

Step 2.13

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9]))$

Step 2.14

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

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9]))$

Step 2.15

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

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9]))$

Step 2.16

Multiply $4$ by $1$ .

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 + \frac{d}{d x} [9]))$

Step 2.17

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

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 + 0))$

Step 2.18

Simplify the expression.

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Step 2.18.1

Add $4$ and $0$ .

$8 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) \cdot 4)$

Step 2.18.2

Multiply $4$ by $- 1$ .

$8 (4 \cos^{2} (4 x + 9) - 4 \sin^{2} (4 x + 9))$

Step 2.19

Simplify.

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Step 2.19.1

Apply the distributive property.

$8 (4 \cos^{2} (4 x + 9)) + 8 (- 4 \sin^{2} (4 x + 9))$

Step 2.19.2

Combine terms.

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Step 2.19.2.1

Multiply $4$ by $8$ .

$32 \cos^{2} (4 x + 9) + 8 (- 4 \sin^{2} (4 x + 9))$

Step 2.19.2.2

Multiply $- 4$ by $8$ .

$f'' (x) = 32 \cos^{2} (4 x + 9) - 32 \sin^{2} (4 x + 9)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $32 \cos^{2} (4 x + 9) - 32 \sin^{2} (4 x + 9)$ with respect to $x$ is $\frac{d}{d x} [32 \cos^{2} (4 x + 9)] + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$ .

$\frac{d}{d x} [32 \cos^{2} (4 x + 9)] + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2

Evaluate $\frac{d}{d x} [32 \cos^{2} (4 x + 9)]$ .

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Step 3.2.1

Since $32$ is constant with respect to $x$ , the derivative of $32 \cos^{2} (4 x + 9)$ with respect to $x$ is $32 \frac{d}{d x} [\cos^{2} (4 x + 9)]$ .

$32 \frac{d}{d x} [\cos^{2} (4 x + 9)] + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (4 x + 9)$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (4 x + 9)$ .

$32 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cos (4 x + 9)]) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.2.2

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

$32 (2 u_{1} \frac{d}{d x} [\cos (4 x + 9)]) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $\cos (4 x + 9)$ .

$32 (2 \cos (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)]) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x + 9$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{2}$ as $4 x + 9$ .

$32 (2 \cos (4 x + 9) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [4 x + 9])) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$32 (2 \cos (4 x + 9) (- \sin (u_{2}) \frac{d}{d x} [4 x + 9])) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.3.3

Replace all occurrences of $u_{2}$ with $4 x + 9$ .

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) \frac{d}{d x} [4 x + 9])) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.4

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9]))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.5

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

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9]))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.6

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

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9]))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.7

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

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) (4 \cdot 1 + 0))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.8

Multiply $4$ by $1$ .

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) (4 + 0))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.9

Add $4$ and $0$ .

$32 (2 \cos (4 x + 9) (- \sin (4 x + 9) \cdot 4)) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.10

Multiply $4$ by $- 1$ .

$32 (2 \cos (4 x + 9) (- 4 \sin (4 x + 9))) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.11

Multiply $- 4$ by $2$ .

$32 (- 8 \cos (4 x + 9) \sin (4 x + 9)) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.2.12

Multiply $- 8$ by $32$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) + \frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$

Step 3.3

Evaluate $\frac{d}{d x} [- 32 \sin^{2} (4 x + 9)]$ .

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Step 3.3.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 \sin^{2} (4 x + 9)$ with respect to $x$ is $- 32 \frac{d}{d x} [\sin^{2} (4 x + 9)]$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 \frac{d}{d x} [\sin^{2} (4 x + 9)]$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (4 x + 9)$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{3}$ as $\sin (4 x + 9)$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\sin (4 x + 9)])$

Step 3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 u_{3} \frac{d}{d x} [\sin (4 x + 9)])$

Step 3.3.2.3

Replace all occurrences of $u_{3}$ with $\sin (4 x + 9)$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) \frac{d}{d x} [\sin (4 x + 9)])$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x + 9$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u_{4}$ as $4 x + 9$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\frac{d}{d u_{4}} [\sin (u_{4})] \frac{d}{d x} [4 x + 9]))$

Step 3.3.3.2

The derivative of $\sin (u_{4})$ with respect to $u_{4}$ is $\cos (u_{4})$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (u_{4}) \frac{d}{d x} [4 x + 9]))$

Step 3.3.3.3

Replace all occurrences of $u_{4}$ with $4 x + 9$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) \frac{d}{d x} [4 x + 9]))$

Step 3.3.4

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9])))$

Step 3.3.5

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

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9])))$

Step 3.3.6

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

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9])))$

Step 3.3.7

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

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) (4 \cdot 1 + 0)))$

Step 3.3.8

Multiply $4$ by $1$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) (4 + 0)))$

Step 3.3.9

Add $4$ and $0$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (\cos (4 x + 9) \cdot 4))$

Step 3.3.10

Move $4$ to the left of $\cos (4 x + 9)$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (2 \sin (4 x + 9) (4 \cdot \cos (4 x + 9)))$

Step 3.3.11

Multiply $4$ by $2$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 32 (8 \sin (4 x + 9) \cos (4 x + 9))$

Step 3.3.12

Multiply $8$ by $- 32$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 256 \sin (4 x + 9) \cos (4 x + 9)$

Step 3.4

Combine terms.

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Step 3.4.1

Reorder the factors of $- 256 \sin (4 x + 9) \cos (4 x + 9)$ .

$- 256 \cos (4 x + 9) \sin (4 x + 9) - 256 \cos (4 x + 9) \sin (4 x + 9)$

Step 3.4.2

Subtract $256 \cos (4 x + 9) \sin (4 x + 9)$ from $- 256 \cos (4 x + 9) \sin (4 x + 9)$ .

$f''' (x) = - 512 \cos (4 x + 9) \sin (4 x + 9)$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 512$ is constant with respect to $x$ , the derivative of $- 512 \cos (4 x + 9) \sin (4 x + 9)$ with respect to $x$ is $- 512 \frac{d}{d x} [\cos (4 x + 9) \sin (4 x + 9)]$ .

$- 512 \frac{d}{d x} [\cos (4 x + 9) \sin (4 x + 9)]$

Step 4.2

$- 512 (\cos (4 x + 9) \frac{d}{d x} [\sin (4 x + 9)] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x + 9$ .

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Step 4.3.1

To apply the Chain Rule, set $u_{1}$ as $4 x + 9$ .

$- 512 (\cos (4 x + 9) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$- 512 (\cos (4 x + 9) (\cos (u_{1}) \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.3.3

Replace all occurrences of $u_{1}$ with $4 x + 9$ .

$- 512 (\cos (4 x + 9) (\cos (4 x + 9) \frac{d}{d x} [4 x + 9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.4

Raise $\cos (4 x + 9)$ to the power of $1$ .

$- 512 (\cos^{1} (4 x + 9) \cos (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.5

Raise $\cos (4 x + 9)$ to the power of $1$ .

$- 512 (\cos^{1} (4 x + 9) \cos^{1} (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.6

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

$- 512 ({\cos (4 x + 9)}^{1 + 1} \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7

Differentiate.

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Step 4.7.1

Add $1$ and $1$ .

$- 512 (\cos^{2} (4 x + 9) \frac{d}{d x} [4 x + 9] + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.2

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$- 512 (\cos^{2} (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.3

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

$- 512 (\cos^{2} (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.4

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

$- 512 (\cos^{2} (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.5

Multiply $4$ by $1$ .

$- 512 (\cos^{2} (4 x + 9) (4 + \frac{d}{d x} [9]) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.6

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

$- 512 (\cos^{2} (4 x + 9) (4 + 0) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.7

Simplify the expression.

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Step 4.7.7.1

Add $4$ and $0$ .

$- 512 (\cos^{2} (4 x + 9) \cdot 4 + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.7.7.2

Move $4$ to the left of $\cos^{2} (4 x + 9)$ .

$- 512 (4 \cdot \cos^{2} (4 x + 9) + \sin (4 x + 9) \frac{d}{d x} [\cos (4 x + 9)])$

Step 4.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x + 9$ .

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Step 4.8.1

To apply the Chain Rule, set $u_{2}$ as $4 x + 9$ .

$- 512 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [4 x + 9]))$

Step 4.8.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$- 512 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (- \sin (u_{2}) \frac{d}{d x} [4 x + 9]))$

Step 4.8.3

Replace all occurrences of $u_{2}$ with $4 x + 9$ .

$- 512 (4 \cos^{2} (4 x + 9) + \sin (4 x + 9) (- \sin (4 x + 9) \frac{d}{d x} [4 x + 9]))$

Step 4.9

Raise $\sin (4 x + 9)$ to the power of $1$ .

$- 512 (4 \cos^{2} (4 x + 9) - (\sin^{1} (4 x + 9) \sin (4 x + 9)) \frac{d}{d x} [4 x + 9])$

Step 4.10

Raise $\sin (4 x + 9)$ to the power of $1$ .

$- 512 (4 \cos^{2} (4 x + 9) - (\sin^{1} (4 x + 9) \sin^{1} (4 x + 9)) \frac{d}{d x} [4 x + 9])$

Step 4.11

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

$- 512 (4 \cos^{2} (4 x + 9) - {\sin (4 x + 9)}^{1 + 1} \frac{d}{d x} [4 x + 9])$

Step 4.12

Add $1$ and $1$ .

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) \frac{d}{d x} [4 x + 9])$

Step 4.13

By the Sum Rule, the derivative of $4 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ .

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [9]))$

Step 4.14

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

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [9]))$

Step 4.15

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

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 \cdot 1 + \frac{d}{d x} [9]))$

Step 4.16

Multiply $4$ by $1$ .

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 + \frac{d}{d x} [9]))$

Step 4.17

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

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) (4 + 0))$

Step 4.18

Simplify the expression.

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Step 4.18.1

Add $4$ and $0$ .

$- 512 (4 \cos^{2} (4 x + 9) - \sin^{2} (4 x + 9) \cdot 4)$

Step 4.18.2

Multiply $4$ by $- 1$ .

$- 512 (4 \cos^{2} (4 x + 9) - 4 \sin^{2} (4 x + 9))$

Step 4.19

Simplify.

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Step 4.19.1

Apply the distributive property.

$- 512 (4 \cos^{2} (4 x + 9)) - 512 (- 4 \sin^{2} (4 x + 9))$

Step 4.19.2

Combine terms.

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Step 4.19.2.1

Multiply $4$ by $- 512$ .

$- 2048 \cos^{2} (4 x + 9) - 512 (- 4 \sin^{2} (4 x + 9))$

Step 4.19.2.2

Multiply $- 4$ by $- 512$ .

$f^{4} (x) = - 2048 \cos^{2} (4 x + 9) + 2048 \sin^{2} (4 x + 9)$