Calculus Examples

$f (x) = \tan (4 x)$

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 $f (x) = \tan (x)$ and $g (x) = 4 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $4 x$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x]$

Step 1.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [4 x]$

Step 1.1.3

Replace all occurrences of $u$ with $4 x$ .

$\sec^{2} (4 x) \frac{d}{d x} [4 x]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\sec^{2} (4 x) (4 \frac{d}{d x} [x])$

Step 1.2.2

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

$\sec^{2} (4 x) (4 \cdot 1)$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $4$ by $1$ .

$\sec^{2} (4 x) \cdot 4$

Step 1.2.3.2

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

$f' (x) = 4 \sec^{2} (4 x)$

Step 2

Find the second derivative.

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Step 2.1

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

$4 \frac{d}{d x} [\sec^{2} (4 x)]$

Step 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 $g (x) = \sec (4 x)$ .

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Step 2.2.1

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

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

Step 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 $n = 2$ .

$4 (2 u_{1} \frac{d}{d x} [\sec (4 x)])$

Step 2.2.3

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

$4 (2 \sec (4 x) \frac{d}{d x} [\sec (4 x)])$

Step 2.3

Multiply $2$ by $4$ .

$8 (\sec (4 x) \frac{d}{d x} [\sec (4 x)])$

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 $f (x) = \sec (x)$ and $g (x) = 4 x$ .

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Step 2.4.1

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

$8 \sec (4 x) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [4 x])$

Step 2.4.2

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

$8 \sec (4 x) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [4 x])$

Step 2.4.3

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

$8 \sec (4 x) (\sec (4 x) \tan (4 x) \frac{d}{d x} [4 x])$

Step 2.5

Raise $\sec (4 x)$ to the power of $1$ .

$8 (\sec^{1} (4 x) \sec (4 x)) (\tan (4 x) \frac{d}{d x} [4 x])$

Step 2.6

Raise $\sec (4 x)$ to the power of $1$ .

$8 (\sec^{1} (4 x) \sec^{1} (4 x)) (\tan (4 x) \frac{d}{d x} [4 x])$

Step 2.7

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

$8 {\sec (4 x)}^{1 + 1} (\tan (4 x) \frac{d}{d x} [4 x])$

Step 2.8

Add $1$ and $1$ .

$8 \sec^{2} (4 x) (\tan (4 x) \frac{d}{d x} [4 x])$

Step 2.9

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

$8 \sec^{2} (4 x) \tan (4 x) (4 \frac{d}{d x} [x])$

Step 2.10

Multiply $4$ by $8$ .

$32 \sec^{2} (4 x) \tan (4 x) \frac{d}{d x} [x]$

Step 2.11

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

$32 \sec^{2} (4 x) \tan (4 x) \cdot 1$

Step 2.12

Multiply $32$ by $1$ .

$f'' (x) = 32 \sec^{2} (4 x) \tan (4 x)$

Step 3

Find the third derivative.

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Step 3.1

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

$32 \frac{d}{d x} [\sec^{2} (4 x) \tan (4 x)]$

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 $f (x) = \sec^{2} (4 x)$ and $g (x) = \tan (4 x)$ .

$32 (\sec^{2} (4 x) \frac{d}{d x} [\tan (4 x)] + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.3

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

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Step 3.3.1

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

$32 (\sec^{2} (4 x) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [4 x]) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.3.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$32 (\sec^{2} (4 x) (\sec^{2} (u_{1}) \frac{d}{d x} [4 x]) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.3.3

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

$32 (\sec^{2} (4 x) (\sec^{2} (4 x) \frac{d}{d x} [4 x]) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.4

Multiply $\sec^{2} (4 x)$ by $\sec^{2} (4 x)$ by adding the exponents.

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Step 3.4.1

Move $\sec^{2} (4 x)$ .

$32 (\sec^{2} (4 x) \sec^{2} (4 x) \frac{d}{d x} [4 x] + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.4.2

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

$32 ({\sec (4 x)}^{2 + 2} \frac{d}{d x} [4 x] + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.4.3

Add $2$ and $2$ .

$32 (\sec^{4} (4 x) \frac{d}{d x} [4 x] + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.5

Differentiate.

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Step 3.5.1

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

$32 (\sec^{4} (4 x) (4 \frac{d}{d x} [x]) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.5.2

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

$32 (\sec^{4} (4 x) (4 \cdot 1) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Multiply $4$ by $1$ .

$32 (\sec^{4} (4 x) \cdot 4 + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.5.3.2

Move $4$ to the left of $\sec^{4} (4 x)$ .

$32 (4 \cdot \sec^{4} (4 x) + \tan (4 x) \frac{d}{d x} [\sec^{2} (4 x)])$

Step 3.6

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 $g (x) = \sec (4 x)$ .

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Step 3.6.1

To apply the Chain Rule, set $u_{2}$ as $\sec (4 x)$ .

$32 (4 \sec^{4} (4 x) + \tan (4 x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sec (4 x)]))$

Step 3.6.2

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

$32 (4 \sec^{4} (4 x) + \tan (4 x) (2 u_{2} \frac{d}{d x} [\sec (4 x)]))$

Step 3.6.3

Replace all occurrences of $u_{2}$ with $\sec (4 x)$ .

$32 (4 \sec^{4} (4 x) + \tan (4 x) (2 \sec (4 x) \frac{d}{d x} [\sec (4 x)]))$

Step 3.7

Move $2$ to the left of $\tan (4 x)$ .

$32 (4 \sec^{4} (4 x) + 2 \cdot \tan (4 x) \sec (4 x) \frac{d}{d x} [\sec (4 x)])$

Step 3.8

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

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Step 3.8.1

To apply the Chain Rule, set $u_{3}$ as $4 x$ .

$32 (4 \sec^{4} (4 x) + 2 \tan (4 x) \sec (4 x) (\frac{d}{d u_{3}} [\sec (u_{3})] \frac{d}{d x} [4 x]))$

Step 3.8.2

The derivative of $\sec (u_{3})$ with respect to $u_{3}$ is $\sec (u_{3}) \tan (u_{3})$ .

$32 (4 \sec^{4} (4 x) + 2 \tan (4 x) \sec (4 x) (\sec (u_{3}) \tan (u_{3}) \frac{d}{d x} [4 x]))$

Step 3.8.3

Replace all occurrences of $u_{3}$ with $4 x$ .

$32 (4 \sec^{4} (4 x) + 2 \tan (4 x) \sec (4 x) (\sec (4 x) \tan (4 x) \frac{d}{d x} [4 x]))$

Step 3.9

Raise $\tan (4 x)$ to the power of $1$ .

$32 (4 \sec^{4} (4 x) + 2 (\tan^{1} (4 x) \tan (4 x)) \sec (4 x) (\sec (4 x) \frac{d}{d x} [4 x]))$

Step 3.10

Raise $\tan (4 x)$ to the power of $1$ .

$32 (4 \sec^{4} (4 x) + 2 (\tan^{1} (4 x) \tan^{1} (4 x)) \sec (4 x) (\sec (4 x) \frac{d}{d x} [4 x]))$

Step 3.11

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

$32 (4 \sec^{4} (4 x) + 2 {\tan (4 x)}^{1 + 1} \sec (4 x) (\sec (4 x) \frac{d}{d x} [4 x]))$

Step 3.12

Add $1$ and $1$ .

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) \sec (4 x) (\sec (4 x) \frac{d}{d x} [4 x]))$

Step 3.13

Raise $\sec (4 x)$ to the power of $1$ .

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) (\sec^{1} (4 x) \sec (4 x)) \frac{d}{d x} [4 x])$

Step 3.14

Raise $\sec (4 x)$ to the power of $1$ .

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) (\sec^{1} (4 x) \sec^{1} (4 x)) \frac{d}{d x} [4 x])$

Step 3.15

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

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) {\sec (4 x)}^{1 + 1} \frac{d}{d x} [4 x])$

Step 3.16

Add $1$ and $1$ .

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) \sec^{2} (4 x) \frac{d}{d x} [4 x])$

Step 3.17

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

$32 (4 \sec^{4} (4 x) + 2 \tan^{2} (4 x) \sec^{2} (4 x) (4 \frac{d}{d x} [x]))$

Step 3.18

Multiply $4$ by $2$ .

$32 (4 \sec^{4} (4 x) + 8 \tan^{2} (4 x) \sec^{2} (4 x) \frac{d}{d x} [x])$

Step 3.19

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

$32 (4 \sec^{4} (4 x) + 8 \tan^{2} (4 x) \sec^{2} (4 x) \cdot 1)$

Step 3.20

Multiply $8$ by $1$ .

$32 (4 \sec^{4} (4 x) + 8 \tan^{2} (4 x) \sec^{2} (4 x))$

Step 3.21

Simplify.

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Step 3.21.1

Apply the distributive property.

$32 (4 \sec^{4} (4 x)) + 32 (8 \tan^{2} (4 x) \sec^{2} (4 x))$

Step 3.21.2

Combine terms.

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Step 3.21.2.1

Multiply $4$ by $32$ .

$128 \sec^{4} (4 x) + 32 (8 \tan^{2} (4 x) \sec^{2} (4 x))$

Step 3.21.2.2

Multiply $8$ by $32$ .

$128 \sec^{4} (4 x) + 256 \tan^{2} (4 x) \sec^{2} (4 x)$

Step 3.21.3

Reorder terms.

$f''' (x) = 256 \sec^{2} (4 x) \tan^{2} (4 x) + 128 \sec^{4} (4 x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $256 \sec^{2} (4 x) \tan^{2} (4 x) + 128 \sec^{4} (4 x)$ with respect to $x$ is $\frac{d}{d x} [256 \sec^{2} (4 x) \tan^{2} (4 x)] + \frac{d}{d x} [128 \sec^{4} (4 x)]$ .

$\frac{d}{d x} [256 \sec^{2} (4 x) \tan^{2} (4 x)] + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2

Evaluate $\frac{d}{d x} [256 \sec^{2} (4 x) \tan^{2} (4 x)]$ .

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Step 4.2.1

Since $256$ is constant with respect to $x$ , the derivative of $256 \sec^{2} (4 x) \tan^{2} (4 x)$ with respect to $x$ is $256 \frac{d}{d x} [\sec^{2} (4 x) \tan^{2} (4 x)]$ .

$256 \frac{d}{d x} [\sec^{2} (4 x) \tan^{2} (4 x)] + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.2

$256 (\sec^{2} (4 x) \frac{d}{d x} [\tan^{2} (4 x)] + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.3

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

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Step 4.2.3.1

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

$256 (\sec^{2} (4 x) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\tan (4 x)]) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.3.2

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

$256 (\sec^{2} (4 x) (2 u_{1} \frac{d}{d x} [\tan (4 x)]) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.3.3

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) \frac{d}{d x} [\tan (4 x)]) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.4

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

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Step 4.2.4.1

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [4 x])) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.4.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (u_{2}) \frac{d}{d x} [4 x])) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.4.3

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) \frac{d}{d x} [4 x])) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.5

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \frac{d}{d x} [x]))) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.6

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) \frac{d}{d x} [\sec^{2} (4 x)]) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.7

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 $g (x) = \sec (4 x)$ .

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Step 4.2.7.1

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\sec (4 x)])) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.7.2

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 u_{3} \frac{d}{d x} [\sec (4 x)])) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.7.3

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) \frac{d}{d x} [\sec (4 x)])) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.8

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

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Step 4.2.8.1

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) (\frac{d}{d u_{4}} [\sec (u_{4})] \frac{d}{d x} [4 x]))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.8.2

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (u_{4}) \tan (u_{4}) \frac{d}{d x} [4 x]))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.8.3

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) \frac{d}{d x} [4 x]))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.9

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \frac{d}{d x} [x])))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.10

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) (4 \cdot 1))) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.11

Multiply $4$ by $1$ .

$256 (\sec^{2} (4 x) (2 \tan (4 x) (\sec^{2} (4 x) \cdot 4)) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.12

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

$256 (\sec^{2} (4 x) (2 \tan (4 x) (4 \cdot \sec^{2} (4 x))) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.13

Multiply $4$ by $2$ .

$256 (\sec^{2} (4 x) (8 \tan (4 x) \sec^{2} (4 x)) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.14

Multiply $\sec^{2} (4 x)$ by $\sec^{2} (4 x)$ by adding the exponents.

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Step 4.2.14.1

Move $\sec^{2} (4 x)$ .

$256 (\sec^{2} (4 x) \sec^{2} (4 x) (8 \tan (4 x)) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.14.2

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

$256 ({\sec (4 x)}^{2 + 2} (8 \tan (4 x)) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.14.3

Add $2$ and $2$ .

$256 (\sec^{4} (4 x) (8 \tan (4 x)) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.15

Move $8$ to the left of $\sec^{4} (4 x)$ .

$256 (8 \cdot \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.16

Multiply $4$ by $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (2 \sec (4 x) (\sec (4 x) \tan (4 x) \cdot 4))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.17

Move $4$ to the left of $\sec (4 x) \tan (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (2 \sec (4 x) (4 \cdot (\sec (4 x) \tan (4 x))))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.18

Multiply $4$ by $2$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (8 \sec (4 x) (\sec (4 x) \tan (4 x)))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.19

Raise $\sec (4 x)$ to the power of $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (8 (\sec^{1} (4 x) \sec (4 x)) \tan (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.20

Raise $\sec (4 x)$ to the power of $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (8 (\sec^{1} (4 x) \sec^{1} (4 x)) \tan (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.21

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

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (8 {\sec (4 x)}^{1 + 1} \tan (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.22

Add $1$ and $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{2} (4 x) (8 \sec^{2} (4 x) \tan (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.23

Multiply $\tan^{2} (4 x)$ by $\tan (4 x)$ by adding the exponents.

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Step 4.2.23.1

Move $\tan (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan (4 x) \tan^{2} (4 x) (8 \sec^{2} (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.23.2

Multiply $\tan (4 x)$ by $\tan^{2} (4 x)$ .

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Step 4.2.23.2.1

Raise $\tan (4 x)$ to the power of $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{1} (4 x) \tan^{2} (4 x) (8 \sec^{2} (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.23.2.2

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

$256 (8 \sec^{4} (4 x) \tan (4 x) + {\tan (4 x)}^{1 + 2} (8 \sec^{2} (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.23.3

Add $1$ and $2$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + \tan^{3} (4 x) (8 \sec^{2} (4 x))) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.2.24

Move $8$ to the left of $\tan^{3} (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + \frac{d}{d x} [128 \sec^{4} (4 x)]$

Step 4.3

Evaluate $\frac{d}{d x} [128 \sec^{4} (4 x)]$ .

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Step 4.3.1

Since $128$ is constant with respect to $x$ , the derivative of $128 \sec^{4} (4 x)$ with respect to $x$ is $128 \frac{d}{d x} [\sec^{4} (4 x)]$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 \frac{d}{d x} [\sec^{4} (4 x)]$

Step 4.3.2

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

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Step 4.3.2.1

To apply the Chain Rule, set $u_{5}$ as $\sec (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (\frac{d}{d u_{5}} [{u_{5}}^{4}] \frac{d}{d x} [\sec (4 x)])$

Step 4.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{5}} [{u_{5}}^{n}]$ is $n {u_{5}}^{n - 1}$ where $n = 4$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 {u_{5}}^{3} \frac{d}{d x} [\sec (4 x)])$

Step 4.3.2.3

Replace all occurrences of $u_{5}$ with $\sec (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) \frac{d}{d x} [\sec (4 x)])$

Step 4.3.3

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

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Step 4.3.3.1

To apply the Chain Rule, set $u_{6}$ as $4 x$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\frac{d}{d u_{6}} [\sec (u_{6})] \frac{d}{d x} [4 x]))$

Step 4.3.3.2

The derivative of $\sec (u_{6})$ with respect to $u_{6}$ is $\sec (u_{6}) \tan (u_{6})$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\sec (u_{6}) \tan (u_{6}) \frac{d}{d x} [4 x]))$

Step 4.3.3.3

Replace all occurrences of $u_{6}$ with $4 x$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\sec (4 x) \tan (4 x) \frac{d}{d x} [4 x]))$

Step 4.3.4

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

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\sec (4 x) \tan (4 x) (4 \frac{d}{d x} [x])))$

Step 4.3.5

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

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\sec (4 x) \tan (4 x) (4 \cdot 1)))$

Step 4.3.6

Multiply $4$ by $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (\sec (4 x) \tan (4 x) \cdot 4))$

Step 4.3.7

Move $4$ to the left of $\sec (4 x) \tan (4 x)$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (4 \sec^{3} (4 x) (4 \cdot (\sec (4 x) \tan (4 x))))$

Step 4.3.8

Multiply $4$ by $4$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (16 \sec^{3} (4 x) (\sec (4 x) \tan (4 x)))$

Step 4.3.9

Raise $\sec (4 x)$ to the power of $1$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (16 (\sec^{1} (4 x) \sec^{3} (4 x)) \tan (4 x))$

Step 4.3.10

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

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (16 {\sec (4 x)}^{1 + 3} \tan (4 x))$

Step 4.3.11

Add $1$ and $3$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 128 (16 \sec^{4} (4 x) \tan (4 x))$

Step 4.3.12

Multiply $16$ by $128$ .

$256 (8 \sec^{4} (4 x) \tan (4 x) + 8 \tan^{3} (4 x) \sec^{2} (4 x)) + 2048 \sec^{4} (4 x) \tan (4 x)$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$256 (8 \sec^{4} (4 x) \tan (4 x)) + 256 (8 \tan^{3} (4 x) \sec^{2} (4 x)) + 2048 \sec^{4} (4 x) \tan (4 x)$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $8$ by $256$ .

$2048 (\sec^{4} (4 x) \tan (4 x)) + 256 (8 \tan^{3} (4 x) \sec^{2} (4 x)) + 2048 \sec^{4} (4 x) \tan (4 x)$

Step 4.4.2.2

Multiply $8$ by $256$ .

$2048 \sec^{4} (4 x) \tan (4 x) + 2048 (\tan^{3} (4 x) \sec^{2} (4 x)) + 2048 \sec^{4} (4 x) \tan (4 x)$

Step 4.4.2.3

Add $2048 \sec^{4} (4 x) \tan (4 x)$ and $2048 \sec^{4} (4 x) \tan (4 x)$ .

$f^{4} (x) = 4096 \sec^{4} (4 x) \tan (4 x) + 2048 \tan^{3} (4 x) \sec^{2} (4 x)$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $4096 \sec^{4} (4 x) \tan (4 x) + 2048 \tan^{3} (4 x) \sec^{2} (4 x)$ .

$4096 \sec^{4} (4 x) \tan (4 x) + 2048 \tan^{3} (4 x) \sec^{2} (4 x)$