Calculus Examples

$\tan (x)$

Step 1

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 2.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$2 \sec (x) (\sec (x) \tan (x))$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

$2 {\sec (x)}^{1 + 1} \tan (x)$

Step 2.6

Add $1$ and $1$ .

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

Step 3

Find the third derivative.

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Step 3.1

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

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

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Add $2$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 3.5.2

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

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

Step 3.5.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 3.6

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

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

Step 3.7

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Add $1$ and $1$ .

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Add $1$ and $1$ .

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

Step 3.16

Simplify.

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Step 3.16.1

Apply the distributive property.

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

Step 3.16.2

Multiply $2$ by $2$ .

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

Step 3.16.3

Reorder terms.

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

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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 $g (x) = \tan (x)$ .

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Step 4.2.3.1

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

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

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 $n = 2$ .

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

Step 4.2.3.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

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

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

Step 4.2.6

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 4.2.7

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

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Step 4.2.7.1

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

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

Step 4.2.7.2

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

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

Step 4.2.7.3

Add $2$ and $2$ .

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

Step 4.2.8

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

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

Step 4.2.9

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

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

Step 4.2.10

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

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

Step 4.2.11

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

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

Step 4.2.12

Add $1$ and $1$ .

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

Step 4.2.13

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

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Step 4.2.13.1

Move $\tan (x)$ .

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

Step 4.2.13.2

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

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Step 4.2.13.2.1

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

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

Step 4.2.13.2.2

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

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

Step 4.2.13.3

Add $1$ and $2$ .

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

Step 4.2.14

Move $2$ to the left of $\tan^{3} (x)$ .

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

Step 4.3

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

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Step 4.3.1

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

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

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

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Step 4.3.2.1

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

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

Step 4.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 4$ .

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

Step 4.3.2.3

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

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

Step 4.3.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 4.3.4

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

$4 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) + 2 (4 (\sec^{1} (x) \sec^{3} (x)) \tan (x))$

Step 4.3.5

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

$4 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) + 2 (4 {\sec (x)}^{1 + 3} \tan (x))$

Step 4.3.6

Add $1$ and $3$ .

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

Step 4.3.7

Multiply $4$ by $2$ .

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

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $2$ by $4$ .

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

Step 4.4.2.2

Multiply $2$ by $4$ .

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

Step 4.4.2.3

Add $8 \sec^{4} (x) \tan (x)$ and $8 \sec^{4} (x) \tan (x)$ .

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