Calculus Examples

$y = 9 \sec (x)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \sec (x)$ with respect to $x$ is $9 \frac{d}{d x} [\sec (x)]$ .

$9 \frac{d}{d x} [\sec (x)]$

Step 1.2

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

$f' (x) = 9 \sec (x) \tan (x)$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \sec (x) \tan (x)$ with respect to $x$ is $9 \frac{d}{d x} [\sec (x) \tan (x)]$ .

$9 \frac{d}{d x} [\sec (x) \tan (x)]$

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

$9 (\sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec (x)])$

Step 2.3

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

$9 (\sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec (x)])$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

Tap for more steps...

Step 2.4.1.1

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

$9 (\sec^{1} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec (x)])$

Step 2.4.1.2

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

$9 ({\sec (x)}^{1 + 2} + \tan (x) \frac{d}{d x} [\sec (x)])$

Step 2.4.2

Add $1$ and $2$ .

$9 (\sec^{3} (x) + \tan (x) \frac{d}{d x} [\sec (x)])$

Step 2.5

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

$9 (\sec^{3} (x) + \tan (x) (\sec (x) \tan (x)))$

Step 2.6

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

$9 (\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x))$

Step 2.7

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

$9 (\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x))$

Step 2.8

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

$9 (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x))$

Step 2.9

Add $1$ and $1$ .

$9 (\sec^{3} (x) + \tan^{2} (x) \sec (x))$

Step 2.10

Simplify.

Tap for more steps...

Step 2.10.1

Apply the distributive property.

$9 \sec^{3} (x) + 9 (\tan^{2} (x) \sec (x))$

Step 2.10.2

Reorder terms.

$f'' (x) = 9 \tan^{2} (x) \sec (x) + 9 \sec^{3} (x)$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $9 \tan^{2} (x) \sec (x) + 9 \sec^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [9 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [9 \sec^{3} (x)]$ .

$\frac{d}{d x} [9 \tan^{2} (x) \sec (x)] + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [9 \tan^{2} (x) \sec (x)]$ .

Tap for more steps...

Step 3.2.1

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

$9 \frac{d}{d x} [\tan^{2} (x) \sec (x)] + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.2

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

$9 (\tan^{2} (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\tan^{2} (x)]) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.3

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

$9 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\tan^{2} (x)]) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.4

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

Tap for more steps...

Step 3.2.4.1

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

$9 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.4.2

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

$9 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 u_{1} \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.4.3

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

$9 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 \tan (x) \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.5

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

$9 (\tan^{2} (x) (\sec (x) \tan (x)) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.6

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

Tap for more steps...

Step 3.2.6.1

Move $\tan (x)$ .

$9 (\tan (x) \tan^{2} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.6.2

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

Tap for more steps...

Step 3.2.6.2.1

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

$9 (\tan^{1} (x) \tan^{2} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.6.2.2

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

$9 ({\tan (x)}^{1 + 2} \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.6.3

Add $1$ and $2$ .

$9 (\tan^{3} (x) \sec (x) + \sec (x) (2 \tan (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.7

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) (\sec^{1} (x) \sec^{2} (x))) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.8

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) {\sec (x)}^{1 + 2}) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.2.9

Add $1$ and $2$ .

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + \frac{d}{d x} [9 \sec^{3} (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [9 \sec^{3} (x)]$ .

Tap for more steps...

Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \sec^{3} (x)$ with respect to $x$ is $9 \frac{d}{d x} [\sec^{3} (x)]$ .

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 \frac{d}{d x} [\sec^{3} (x)]$

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 $f (x) = x^{3}$ and $g (x) = \sec (x)$ .

Tap for more steps...

Step 3.3.2.1

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\sec (x)])$

Step 3.3.2.2

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 {u_{2}}^{2} \frac{d}{d x} [\sec (x)])$

Step 3.3.2.3

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)])$

Step 3.3.3

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 \sec^{2} (x) (\sec (x) \tan (x)))$

Step 3.3.4

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 (\sec^{1} (x) \sec^{2} (x)) \tan (x))$

Step 3.3.5

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

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 {\sec (x)}^{1 + 2} \tan (x))$

Step 3.3.6

Add $1$ and $2$ .

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 9 (3 \sec^{3} (x) \tan (x))$

Step 3.3.7

Multiply $3$ by $9$ .

$9 (\tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{3} (x)) + 27 \sec^{3} (x) \tan (x)$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

$9 (\tan^{3} (x) \sec (x)) + 9 (2 \tan (x) \sec^{3} (x)) + 27 \sec^{3} (x) \tan (x)$

Step 3.4.2

Combine terms.

Tap for more steps...

Step 3.4.2.1

Multiply $2$ by $9$ .

$9 \tan^{3} (x) \sec (x) + 18 (\tan (x) \sec^{3} (x)) + 27 \sec^{3} (x) \tan (x)$

Step 3.4.2.2

Reorder the factors of $18 \tan (x) \sec^{3} (x)$ .

$9 \tan^{3} (x) \sec (x) + 18 \sec^{3} (x) \tan (x) + 27 \sec^{3} (x) \tan (x)$

Step 3.4.2.3

Add $18 \sec^{3} (x) \tan (x)$ and $27 \sec^{3} (x) \tan (x)$ .

$f''' (x) = 9 \tan^{3} (x) \sec (x) + 45 \sec^{3} (x) \tan (x)$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $9 \tan^{3} (x) \sec (x) + 45 \sec^{3} (x) \tan (x)$ with respect to $x$ is $\frac{d}{d x} [9 \tan^{3} (x) \sec (x)] + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$ .

$\frac{d}{d x} [9 \tan^{3} (x) \sec (x)] + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2

Evaluate $\frac{d}{d x} [9 \tan^{3} (x) \sec (x)]$ .

Tap for more steps...

Step 4.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \tan^{3} (x) \sec (x)$ with respect to $x$ is $9 \frac{d}{d x} [\tan^{3} (x) \sec (x)]$ .

$9 \frac{d}{d x} [\tan^{3} (x) \sec (x)] + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.2

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 $f (x) = \tan^{3} (x)$ and $g (x) = \sec (x)$ .

$9 (\tan^{3} (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\tan^{3} (x)]) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.3

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

$9 (\tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\tan^{3} (x)]) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

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 $f (x) = x^{3}$ and $g (x) = \tan (x)$ .

Tap for more steps...

Step 4.2.4.1

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

$9 (\tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.4.2

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

$9 (\tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (3 {u_{1}}^{2} \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.4.3

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

$9 (\tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (3 \tan^{2} (x) \frac{d}{d x} [\tan (x)])) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.5

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

$9 (\tan^{3} (x) (\sec (x) \tan (x)) + \sec (x) (3 \tan^{2} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.6

Multiply $\tan^{3} (x)$ by $\tan (x)$ by adding the exponents.

Tap for more steps...

Step 4.2.6.1

Move $\tan (x)$ .

$9 (\tan (x) \tan^{3} (x) \sec (x) + \sec (x) (3 \tan^{2} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.6.2

Multiply $\tan (x)$ by $\tan^{3} (x)$ .

Tap for more steps...

Step 4.2.6.2.1

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

$9 (\tan^{1} (x) \tan^{3} (x) \sec (x) + \sec (x) (3 \tan^{2} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.6.2.2

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

$9 ({\tan (x)}^{1 + 3} \sec (x) + \sec (x) (3 \tan^{2} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.6.3

Add $1$ and $3$ .

$9 (\tan^{4} (x) \sec (x) + \sec (x) (3 \tan^{2} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.7

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) (\sec^{1} (x) \sec^{2} (x))) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.8

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) {\sec (x)}^{1 + 2}) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.2.9

Add $1$ and $2$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + \frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$

Step 4.3

Evaluate $\frac{d}{d x} [45 \sec^{3} (x) \tan (x)]$ .

Tap for more steps...

Step 4.3.1

Since $45$ is constant with respect to $x$ , the derivative of $45 \sec^{3} (x) \tan (x)$ with respect to $x$ is $45 \frac{d}{d x} [\sec^{3} (x) \tan (x)]$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 \frac{d}{d x} [\sec^{3} (x) \tan (x)]$

Step 4.3.2

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 $f (x) = \sec^{3} (x)$ and $g (x) = \tan (x)$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec^{3} (x)])$

Step 4.3.3

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec^{3} (x)])$

Step 4.3.4

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

Tap for more steps...

Step 4.3.4.1

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \sec^{2} (x) + \tan (x) (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\sec (x)]))$

Step 4.3.4.2

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \sec^{2} (x) + \tan (x) (3 {u_{2}}^{2} \frac{d}{d x} [\sec (x)]))$

Step 4.3.4.3

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \sec^{2} (x) + \tan (x) (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)]))$

Step 4.3.5

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{3} (x) \sec^{2} (x) + \tan (x) (3 \sec^{2} (x) (\sec (x) \tan (x))))$

Step 4.3.6

Multiply $\sec^{3} (x)$ by $\sec^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 4.3.6.1

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 ({\sec (x)}^{3 + 2} + \tan (x) (3 \sec^{2} (x) (\sec (x) \tan (x))))$

Step 4.3.6.2

Add $3$ and $2$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + \tan (x) (3 \sec^{2} (x) (\sec (x) \tan (x))))$

Step 4.3.7

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + \tan (x) (3 (\sec^{1} (x) \sec^{2} (x)) \tan (x)))$

Step 4.3.8

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + \tan (x) (3 {\sec (x)}^{1 + 2} \tan (x)))$

Step 4.3.9

Add $1$ and $2$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + \tan (x) (3 \sec^{3} (x) \tan (x)))$

Step 4.3.10

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + 3 \sec^{3} (x) (\tan^{1} (x) \tan (x)))$

Step 4.3.11

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + 3 \sec^{3} (x) (\tan^{1} (x) \tan^{1} (x)))$

Step 4.3.12

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

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + 3 \sec^{3} (x) {\tan (x)}^{1 + 1})$

Step 4.3.13

Add $1$ and $1$ .

$9 (\tan^{4} (x) \sec (x) + 3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + 3 \sec^{3} (x) \tan^{2} (x))$

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Apply the distributive property.

$9 (\tan^{4} (x) \sec (x)) + 9 (3 \tan^{2} (x) \sec^{3} (x)) + 45 (\sec^{5} (x) + 3 \sec^{3} (x) \tan^{2} (x))$

Step 4.4.2

Apply the distributive property.

$9 (\tan^{4} (x) \sec (x)) + 9 (3 \tan^{2} (x) \sec^{3} (x)) + 45 \sec^{5} (x) + 45 (3 \sec^{3} (x) \tan^{2} (x))$

Step 4.4.3

Combine terms.

Tap for more steps...

Step 4.4.3.1

Multiply $3$ by $9$ .

$9 \tan^{4} (x) \sec (x) + 27 (\tan^{2} (x) \sec^{3} (x)) + 45 \sec^{5} (x) + 45 (3 \sec^{3} (x) \tan^{2} (x))$

Step 4.4.3.2

Multiply $3$ by $45$ .

$9 \tan^{4} (x) \sec (x) + 27 \tan^{2} (x) \sec^{3} (x) + 45 \sec^{5} (x) + 135 (\sec^{3} (x) \tan^{2} (x))$

Step 4.4.3.3

Reorder the factors of $27 \tan^{2} (x) \sec^{3} (x)$ .

$9 \tan^{4} (x) \sec (x) + 27 \sec^{3} (x) \tan^{2} (x) + 45 \sec^{5} (x) + 135 \sec^{3} (x) \tan^{2} (x)$

Step 4.4.3.4

Add $27 \sec^{3} (x) \tan^{2} (x)$ and $135 \sec^{3} (x) \tan^{2} (x)$ .

$f^{4} (x) = 9 \tan^{4} (x) \sec (x) + 162 \sec^{3} (x) \tan^{2} (x) + 45 \sec^{5} (x)$