Calculus Examples

$\sec (x - \frac{π}{2})$

Step 1

Find the first derivative.

Tap for more steps...

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 $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 1.1.1

To apply the Chain Rule, set $u$ as $x - \frac{π}{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $x - \frac{π}{2}$ .

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)$

Step 1.2.4

Simplify the expression.

Tap for more steps...

Step 1.2.4.1

Add $1$ and $0$ .

$\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \cdot 1$

Step 1.2.4.2

Multiply $\sec (x - \frac{π}{2})$ by $1$ .

$f' (x) = \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})$

Step 2

Find the second derivative.

Tap for more steps...

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 $f (x) = \sec (x - \frac{π}{2})$ and $g (x) = \tan (x - \frac{π}{2})$ .

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

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 $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x - \frac{π}{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x - \frac{π}{2}$ .

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

Step 2.3

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

Add $2$ and $1$ .

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

Step 2.5.2

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

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

Step 2.5.3

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

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

Step 2.5.4

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\sec^{3} (x - \frac{π}{2}) (1 + 0) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})]$

Step 2.5.5

Simplify the expression.

Tap for more steps...

Step 2.5.5.1

Add $1$ and $0$ .

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

Step 2.5.5.2

Multiply $\sec^{3} (x - \frac{π}{2})$ by $1$ .

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

Step 2.6

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

Tap for more steps...

Step 2.6.1

To apply the Chain Rule, set $u_{2}$ as $x - \frac{π}{2}$ .

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

Step 2.6.2

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

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

Step 2.6.3

Replace all occurrences of $u_{2}$ with $x - \frac{π}{2}$ .

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

Step 2.7

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.8

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.9

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

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

Step 2.10

Add $1$ and $1$ .

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

Step 2.11

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

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

Step 2.12

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

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

Step 2.13

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\sec^{3} (x - \frac{π}{2}) + \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) (1 + 0)$

Step 2.14

Simplify the expression.

Tap for more steps...

Step 2.14.1

Add $1$ and $0$ .

$\sec^{3} (x - \frac{π}{2}) + \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) \cdot 1$

Step 2.14.2

Multiply $\tan^{2} (x - \frac{π}{2})$ by $1$ .

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

Step 2.14.3

Reorder terms.

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

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

Tap for more steps...

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

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

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $x - \frac{π}{2}$ .

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $x - \frac{π}{2}$ .

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

Step 3.2.3

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

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

Step 3.2.4

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

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

Step 3.2.5

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{2} (x - \frac{π}{2})] + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.6

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 $g (x) = \tan (x - \frac{π}{2})$ .

Tap for more steps...

Step 3.2.6.1

To apply the Chain Rule, set $u_{2}$ as $\tan (x - \frac{π}{2})$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.6.2

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

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 u_{2} \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.6.3

Replace all occurrences of $u_{2}$ with $\tan (x - \frac{π}{2})$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.7

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

Tap for more steps...

Step 3.2.7.1

To apply the Chain Rule, set $u_{3}$ as $x - \frac{π}{2}$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\frac{d}{d u_{3}} [\tan (u_{3})] \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.7.2

The derivative of $\tan (u_{3})$ with respect to $u_{3}$ is $\sec^{2} (u_{3})$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (u_{3}) \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.7.3

Replace all occurrences of $u_{3}$ with $x - \frac{π}{2}$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.8

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.9

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

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.10

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.11

Add $1$ and $0$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \cdot 1) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.12

Multiply $\sec (x - \frac{π}{2})$ by $1$ .

$\tan^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.13

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

Tap for more steps...

Step 3.2.13.1

Move $\tan (x - \frac{π}{2})$ .

$\tan (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.13.2

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

Tap for more steps...

Step 3.2.13.2.1

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

$\tan^{1} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.13.2.2

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

${\tan (x - \frac{π}{2})}^{1 + 2} \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.13.3

Add $1$ and $2$ .

$\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (2 \tan (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$

Step 3.2.14

Add $1$ and $0$ .

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

Step 3.2.15

Multiply $\sec^{2} (x - \frac{π}{2})$ by $1$ .

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

Step 3.2.16

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

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

Step 3.2.17

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

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

Step 3.2.18

Add $1$ and $2$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [\sec^{3} (x - \frac{π}{2})]$ .

Tap for more steps...

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 $g (x) = \sec (x - \frac{π}{2})$ .

Tap for more steps...

Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

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 $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 3.3.2.1

To apply the Chain Rule, set $u_{5}$ as $x - \frac{π}{2}$ .

$\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 2 \tan (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 3 \sec^{2} (x - \frac{π}{2}) (\frac{d}{d u_{5}} [\sec (u_{5})] \frac{d}{d x} [x - \frac{π}{2}])$

Step 3.3.2.2

The derivative of $\sec (u_{5})$ with respect to $u_{5}$ is $\sec (u_{5}) \tan (u_{5})$ .

$\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 2 \tan (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 3 \sec^{2} (x - \frac{π}{2}) (\sec (u_{5}) \tan (u_{5}) \frac{d}{d x} [x - \frac{π}{2}])$

Step 3.3.2.3

Replace all occurrences of $u_{5}$ with $x - \frac{π}{2}$ .

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

Step 3.3.3

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

Step 3.3.4

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

Step 3.3.5

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 2 \tan (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0))$

Step 3.3.6

Add $1$ and $0$ .

$\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 2 \tan (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \cdot 1)$

Step 3.3.7

Multiply $\sec (x - \frac{π}{2})$ by $1$ .

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

Step 3.3.8

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

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

Step 3.3.9

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

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

Step 3.3.10

Add $1$ and $2$ .

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

Step 3.4

Combine terms.

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

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

$f''' (x) = \tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})$ with respect to $x$ is $\frac{d}{d x} [\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$ .

$\frac{d}{d x} [\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2

Evaluate $\frac{d}{d x} [\tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2})]$ .

Tap for more steps...

Step 4.2.1

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 $f (x) = \tan^{3} (x - \frac{π}{2})$ and $g (x) = \sec (x - \frac{π}{2})$ .

$\tan^{3} (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})] + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

To apply the Chain Rule, set $u_{1}$ as $x - \frac{π}{2}$ .

$\tan^{3} (x - \frac{π}{2}) (\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [x - \frac{π}{2}]) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.2.2

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

$\tan^{3} (x - \frac{π}{2}) (\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [x - \frac{π}{2}]) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.2.3

Replace all occurrences of $u_{1}$ with $x - \frac{π}{2}$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}]) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.3

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}])) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.4

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

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + \frac{d}{d x} [- \frac{π}{2}])) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.5

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) \frac{d}{d x} [\tan^{3} (x - \frac{π}{2})] + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \tan (x - \frac{π}{2})$ .

Tap for more steps...

Step 4.2.6.1

To apply the Chain Rule, set $u_{2}$ as $\tan (x - \frac{π}{2})$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.6.2

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

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 {u_{2}}^{2} \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.6.3

Replace all occurrences of $u_{2}$ with $\tan (x - \frac{π}{2})$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) \frac{d}{d x} [\tan (x - \frac{π}{2})]) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

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 $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 4.2.7.1

To apply the Chain Rule, set $u_{3}$ as $x - \frac{π}{2}$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\frac{d}{d u_{3}} [\tan (u_{3})] \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.7.2

The derivative of $\tan (u_{3})$ with respect to $u_{3}$ is $\sec^{2} (u_{3})$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (u_{3}) \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.7.3

Replace all occurrences of $u_{3}$ with $x - \frac{π}{2}$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}])) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.8

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.9

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

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.10

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.11

Add $1$ and $0$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \cdot 1) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.12

Multiply $\sec (x - \frac{π}{2})$ by $1$ .

$\tan^{3} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.13

Multiply $\tan^{3} (x - \frac{π}{2})$ by $\tan (x - \frac{π}{2})$ by adding the exponents.

Tap for more steps...

Step 4.2.13.1

Move $\tan (x - \frac{π}{2})$ .

$\tan (x - \frac{π}{2}) \tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.13.2

Multiply $\tan (x - \frac{π}{2})$ by $\tan^{3} (x - \frac{π}{2})$ .

Tap for more steps...

Step 4.2.13.2.1

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

$\tan^{1} (x - \frac{π}{2}) \tan^{3} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.13.2.2

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

${\tan (x - \frac{π}{2})}^{1 + 3} \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.13.3

Add $1$ and $3$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0))) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.14

Add $1$ and $0$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \cdot 1)) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.15

Multiply $\sec^{2} (x - \frac{π}{2})$ by $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + \sec (x - \frac{π}{2}) (3 \tan^{2} (x - \frac{π}{2}) \sec^{2} (x - \frac{π}{2})) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.16

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) (\sec^{1} (x - \frac{π}{2}) \sec^{2} (x - \frac{π}{2})) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.17

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) {\sec (x - \frac{π}{2})}^{1 + 2} + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.2.18

Add $1$ and $2$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + \frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 4.3

Evaluate $\frac{d}{d x} [5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$ .

Tap for more steps...

Step 4.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})$ with respect to $x$ is $5 \frac{d}{d x} [\sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 \frac{d}{d x} [\sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

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 $f (x) = \sec^{3} (x - \frac{π}{2})$ and $g (x) = \tan (x - \frac{π}{2})$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) \frac{d}{d x} [\tan (x - \frac{π}{2})] + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})])$

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 $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 4.3.3.1

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

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

Step 4.3.3.2

The derivative of $\tan (u_{4})$ with respect to $u_{4}$ is $\sec^{2} (u_{4})$ .

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

Step 4.3.3.3

Replace all occurrences of $u_{4}$ with $x - \frac{π}{2}$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}]) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})])$

Step 4.3.4

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}])) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})])$

Step 4.3.5

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + \frac{d}{d x} [- \frac{π}{2}])) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})])$

Step 4.3.6

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) (1 + 0)) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec^{3} (x - \frac{π}{2})])$

Step 4.3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sec (x - \frac{π}{2})$ .

Tap for more steps...

Step 4.3.7.1

To apply the Chain Rule, set $u_{5}$ as $\sec (x - \frac{π}{2})$ .

Step 4.3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{5}} [{u_{5}}^{n}]$ is $n {u_{5}}^{n - 1}$ where $n = 3$ .

Step 4.3.7.3

Replace all occurrences of $u_{5}$ with $\sec (x - \frac{π}{2})$ .

Step 4.3.8

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 $g (x) = x - \frac{π}{2}$ .

Tap for more steps...

Step 4.3.8.1

To apply the Chain Rule, set $u_{6}$ as $x - \frac{π}{2}$ .

Step 4.3.8.2

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

Step 4.3.8.3

Replace all occurrences of $u_{6}$ with $x - \frac{π}{2}$ .

Step 4.3.9

By the Sum Rule, the derivative of $x - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]$ .

Step 4.3.10

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

Step 4.3.11

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

Step 4.3.12

Add $1$ and $0$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \cdot 1) + \tan (x - \frac{π}{2}) (3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0))))$

Step 4.3.13

Multiply $\sec^{2} (x - \frac{π}{2})$ by $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{3} (x - \frac{π}{2}) \sec^{2} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0))))$

Step 4.3.14

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

Tap for more steps...

Step 4.3.14.1

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 ({\sec (x - \frac{π}{2})}^{3 + 2} + \tan (x - \frac{π}{2}) (3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0))))$

Step 4.3.14.2

Add $3$ and $2$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (3 \sec^{2} (x - \frac{π}{2}) (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0))))$

Step 4.3.15

Add $1$ and $0$ .

Step 4.3.16

Multiply $\sec (x - \frac{π}{2})$ by $1$ .

Step 4.3.17

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (3 (\sec^{1} (x - \frac{π}{2}) \sec^{2} (x - \frac{π}{2})) \tan (x - \frac{π}{2})))$

Step 4.3.18

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (3 {\sec (x - \frac{π}{2})}^{1 + 2} \tan (x - \frac{π}{2})))$

Step 4.3.19

Add $1$ and $2$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (3 \sec^{3} (x - \frac{π}{2}) \tan (x - \frac{π}{2})))$

Step 4.3.20

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + 3 \sec^{3} (x - \frac{π}{2}) (\tan^{1} (x - \frac{π}{2}) \tan (x - \frac{π}{2})))$

Step 4.3.21

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

Step 4.3.22

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + 3 \sec^{3} (x - \frac{π}{2}) {\tan (x - \frac{π}{2})}^{1 + 1})$

Step 4.3.23

Add $1$ and $1$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 (\sec^{5} (x - \frac{π}{2}) + 3 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}))$

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Apply the distributive property.

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 \sec^{5} (x - \frac{π}{2}) + 5 (3 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}))$

Step 4.4.2

Combine terms.

Tap for more steps...

Step 4.4.2.1

Multiply $3$ by $5$ .

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \tan^{2} (x - \frac{π}{2}) \sec^{3} (x - \frac{π}{2}) + 5 \sec^{5} (x - \frac{π}{2}) + 15 (\sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}))$

Step 4.4.2.2

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

$\tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 3 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}) + 5 \sec^{5} (x - \frac{π}{2}) + 15 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2})$

Step 4.4.2.3

Add $3 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2})$ and $15 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2})$ .

$f^{4} (x) = \tan^{4} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 18 \sec^{3} (x - \frac{π}{2}) \tan^{2} (x - \frac{π}{2}) + 5 \sec^{5} (x - \frac{π}{2})$