Algebra Examples

$\sec (x)$

Step 1

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

$f' (x) = \sec (x) \tan (x)$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

Add $1$ and $2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $1$ and $1$ .

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

Step 2.9

Reorder terms.

$f'' (x) = \tan^{2} (x) \sec (x) + \sec^{3} (x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\tan^{2} (x) \sec (x) + \sec^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [\tan^{2} (x) \sec (x)] + \frac{d}{d x} [\sec^{3} (x)]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

Move $\tan (x)$ .

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

$\tan^{1} (x) \tan^{2} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x)) + \frac{d}{d x} [\sec^{3} (x)]$

Step 3.2.5.2.2

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

${\tan (x)}^{1 + 2} \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x)) + \frac{d}{d x} [\sec^{3} (x)]$

Step 3.2.5.3

Add $1$ and $2$ .

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

Step 3.2.6

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

$\tan^{3} (x) \sec (x) + 2 \tan (x) (\sec^{1} (x) \sec^{2} (x)) + \frac{d}{d x} [\sec^{3} (x)]$

Step 3.2.7

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

$\tan^{3} (x) \sec (x) + 2 \tan (x) {\sec (x)}^{1 + 2} + \frac{d}{d x} [\sec^{3} (x)]$

Step 3.2.8

Add $1$ and $2$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [\sec^{3} (x)]$ .

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Step 3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sec (x)$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $2$ .

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

Step 3.4

Combine terms.

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Step 3.4.1

Reorder the factors of $2 \tan (x) \sec^{3} (x)$ .

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

Step 3.4.2

Add $2 \sec^{3} (x) \tan (x)$ and $3 \sec^{3} (x) \tan (x)$ .

$f''' (x) = \tan^{3} (x) \sec (x) + 5 \sec^{3} (x) \tan (x)$