Calculus Examples

$f (θ) = \sec^{2} (4 θ)$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

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

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

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 $n = 2$ .

$2 u_{1} \frac{d}{d θ} [\sec (4 θ)]$

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 1.8

Multiply $4$ by $2$ .

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

Step 1.9

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

$8 \sec^{2} (4 θ) \tan (4 θ) \cdot 1$

Step 1.10

Multiply $8$ by $1$ .

$f' (θ) = 8 \sec^{2} (4 θ) \tan (4 θ)$

Step 2

Find the second derivative.

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Step 2.1

Since $8$ is constant with respect to $θ$ , the derivative of $8 \sec^{2} (4 θ) \tan (4 θ)$ with respect to $θ$ is $8 \frac{d}{d θ} [\sec^{2} (4 θ) \tan (4 θ)]$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = \sec^{2} (4 θ)$ and $g (θ) = \tan (4 θ)$ .

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

Step 2.3

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

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Step 2.3.1

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

$8 (\sec^{2} (4 θ) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d θ} [4 θ]) + \tan (4 θ) \frac{d}{d θ} [\sec^{2} (4 θ)])$

Step 2.3.2

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

$8 (\sec^{2} (4 θ) (\sec^{2} (u_{1}) \frac{d}{d θ} [4 θ]) + \tan (4 θ) \frac{d}{d θ} [\sec^{2} (4 θ)])$

Step 2.3.3

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

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

Step 2.4

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

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Step 2.4.1

Move $\sec^{2} (4 θ)$ .

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

Step 2.4.2

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

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

Step 2.4.3

Add $2$ and $2$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Simplify the expression.

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Step 2.5.3.1

Multiply $4$ by $1$ .

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

Step 2.5.3.2

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

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

$8 (4 \sec^{4} (4 θ) + 2 \tan (4 θ) \sec (4 θ) (\frac{d}{d u_{3}} [\sec (u_{3})] \frac{d}{d θ} [4 θ]))$

Step 2.8.2

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

$8 (4 \sec^{4} (4 θ) + 2 \tan (4 θ) \sec (4 θ) (\sec (u_{3}) \tan (u_{3}) \frac{d}{d θ} [4 θ]))$

Step 2.8.3

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Add $1$ and $1$ .

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

Step 2.17

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

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

Step 2.18

Multiply $4$ by $2$ .

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

Step 2.19

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

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

Step 2.20

Multiply $8$ by $1$ .

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

Step 2.21

Simplify.

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Step 2.21.1

Apply the distributive property.

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

Step 2.21.2

Combine terms.

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Step 2.21.2.1

Multiply $4$ by $8$ .

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

Step 2.21.2.2

Multiply $8$ by $8$ .

$32 \sec^{4} (4 θ) + 64 \tan^{2} (4 θ) \sec^{2} (4 θ)$

Step 2.21.3

Reorder terms.

$f'' (θ) = 64 \sec^{2} (4 θ) \tan^{2} (4 θ) + 32 \sec^{4} (4 θ)$

Step 3

The second derivative of $f (θ)$ with respect to $θ$ is $64 \sec^{2} (4 θ) \tan^{2} (4 θ) + 32 \sec^{4} (4 θ)$ .

$64 \sec^{2} (4 θ) \tan^{2} (4 θ) + 32 \sec^{4} (4 θ)$