Calculus Examples

$y = arcsec (x) - 6 x$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $arcsec (x) - 6 x$ with respect to $x$ is $\frac{d}{d x} [arcsec (x)] + \frac{d}{d x} [- 6 x]$ .

$\frac{d}{d x} [arcsec (x)] + \frac{d}{d x} [- 6 x]$

Step 1.2

The derivative of $arcsec (x)$ with respect to $x$ is $\frac{1}{x \sqrt{x^{2} - 1}}$ .

$\frac{1}{x \sqrt{x^{2} - 1}} + \frac{d}{d x} [- 6 x]$

Step 1.3

Evaluate $\frac{d}{d x} [- 6 x]$ .

Tap for more steps...

Step 1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{1}{x \sqrt{x^{2} - 1}} - 6 \frac{d}{d x} [x]$

Step 1.3.2

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

$\frac{1}{x \sqrt{x^{2} - 1}} - 6 \cdot 1$

Step 1.3.3

Multiply $- 6$ by $1$ .

$\frac{1}{x \sqrt{x^{2} - 1}} - 6$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $\frac{1}{x \sqrt{x^{2} - 1}} - 6$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x \sqrt{x^{2} - 1}}] + \frac{d}{d x} [- 6]$ .

$f'' (x) = \frac{d}{d x} (\frac{1}{x \sqrt{x^{2} - 1}}) + \frac{d}{d x} (- 6)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{1}{x \sqrt{x^{2} - 1}}]$ .

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 1}$ as ${(x^{2} - 1)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{d}{d x} (\frac{1}{x {(x^{2} - 1)}^{\frac{1}{2}}}) + \frac{d}{d x} (- 6)$

Step 2.2.2

Rewrite $\frac{1}{x {(x^{2} - 1)}^{\frac{1}{2}}}$ as ${(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 1}$ .

$f'' (x) = \frac{d}{d x} ({(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 1}) + \frac{d}{d x} (- 6)$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x {(x^{2} - 1)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.2.3.1

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

$f'' (x) = \frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x {(x^{2} - 1)}^{\frac{1}{2}}) + \frac{d}{d x} (- 6)$

Step 2.2.3.2

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

$f'' (x) = - {u_{1}}^{- 2} \frac{d}{d x} (x {(x^{2} - 1)}^{\frac{1}{2}}) + \frac{d}{d x} (- 6)$

Step 2.2.3.3

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{d}{d x} (x {(x^{2} - 1)}^{\frac{1}{2}}) + \frac{d}{d x} (- 6)$

Step 2.2.4

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 $f (x) = x$ and $g (x) = {(x^{2} - 1)}^{\frac{1}{2}}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x \frac{d}{d x} ({(x^{2} - 1)}^{\frac{1}{2}}) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} - 1$ .

Tap for more steps...

Step 2.2.5.1

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{d}{d u (2)} ({u_{2}}^{\frac{1}{2}}) \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.5.2

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.5.3

Replace all occurrences of $u_{2}$ with $x^{2} - 1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot {(x^{2} - 1)}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.6

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} - 1} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1)))) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.7

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} - 1} (2 x + \frac{d}{d x} (- 1)))) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} - 1} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6)$

Step 2.2.9

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} - 1} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.11

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.12

Combine the numerators over the common denominator.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1 - 1 \cdot 2}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.13

Simplify the numerator.

Tap for more steps...

Step 2.2.13.1

Multiply $- 1$ by $2$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{1 - 2}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.13.2

Subtract $2$ from $1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{\frac{- 1}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.14

Move the negative in front of the fraction.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{- \frac{1}{2}} (2 x + 0))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.15

Add $2 x$ and $0$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{1}{2} \cdot ({(x^{2} - 1)}^{- \frac{1}{2}} (2 x))) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.16

Combine $\frac{1}{2}$ and ${(x^{2} - 1)}^{- \frac{1}{2}}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{{(x^{2} - 1)}^{- \frac{1}{2}}}{2} \cdot (2 x)) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.17

Combine $2$ and $\frac{{(x^{2} - 1)}^{- \frac{1}{2}}}{2}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{2 {(x^{2} - 1)}^{- \frac{1}{2}}}{2} \cdot x) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.18

Combine $\frac{2 {(x^{2} - 1)}^{- \frac{1}{2}}}{2}$ and $x$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{2 {(x^{2} - 1)}^{- \frac{1}{2}} x}{2}) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.19

Move ${(x^{2} - 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{2 x}{2 {(x^{2} - 1)}^{\frac{1}{2}}}) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.20

Cancel the common factor.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{2 x}{2 {(x^{2} - 1)}^{\frac{1}{2}}}) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.21

Rewrite the expression.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (x (\frac{x}{{(x^{2} - 1)}^{\frac{1}{2}}}) + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.22

Combine $x$ and $\frac{x}{{(x^{2} - 1)}^{\frac{1}{2}}}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x \cdot x}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.23

Raise $x$ to the power of $1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x \cdot x}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.24

Raise $x$ to the power of $1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x \cdot x}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.25

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x^{1 + 1}}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.26

Add $1$ and $1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x^{2}}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (- 6)$

Step 2.2.27

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x^{2}}{{(x^{2} - 1)}^{\frac{1}{2}}} + {(x^{2} - 1)}^{\frac{1}{2}}) + \frac{d}{d x} (- 6)$

Step 2.2.28

To write ${(x^{2} - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - 1)}^{\frac{1}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} (\frac{x^{2}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{{(x^{2} - 1)}^{\frac{1}{2}} {(x^{2} - 1)}^{\frac{1}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}}) + \frac{d}{d x} (- 6)$

Step 2.2.29

Combine the numerators over the common denominator.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + {(x^{2} - 1)}^{\frac{1}{2}} {(x^{2} - 1)}^{\frac{1}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.30

Multiply ${(x^{2} - 1)}^{\frac{1}{2}}$ by ${(x^{2} - 1)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 2.2.30.1

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

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + {(x^{2} - 1)}^{\frac{1}{2} + \frac{1}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.30.2

Combine the numerators over the common denominator.

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + {(x^{2} - 1)}^{\frac{1 + 1}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.30.3

Add $1$ and $1$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + {(x^{2} - 1)}^{\frac{2}{2}}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.30.4

Divide $2$ by $2$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.31

Simplify ${(x^{2} - 1)}^{1}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{x^{2} + x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.32

Add $x^{2}$ and $x^{2}$ .

$f'' (x) = - {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2} \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.33

Combine $\frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}}}$ and ${(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2}$ .

$f'' (x) = - \frac{(2 x^{2} - 1) {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2}}{{(x^{2} - 1)}^{\frac{1}{2}}} + \frac{d}{d x} (- 6)$

Step 2.2.34

Move ${(x {(x^{2} - 1)}^{\frac{1}{2}})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (- 6)$

Step 2.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} {(x {(x^{2} - 1)}^{\frac{1}{2}})}^{2}} + 0$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the product rule to $x {(x^{2} - 1)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} (x^{2} {({(x^{2} - 1)}^{\frac{1}{2}})}^{2})} + 0$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Multiply the exponents in ${({(x^{2} - 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.4.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} (x^{2} {(x^{2} - 1)}^{\frac{1}{2} \cdot 2})} + 0$

Step 2.4.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.1.2.1

Cancel the common factor.

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} (x^{2} {(x^{2} - 1)}^{\frac{1}{2} \cdot 2})} + 0$

Step 2.4.2.1.2.2

Rewrite the expression.

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} (x^{2} (x^{2} - 1))} + 0$

Step 2.4.2.2

Simplify.

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{1}{2}} (x^{2} (x^{2} - 1))} + 0$

Step 2.4.2.3

Multiply ${(x^{2} - 1)}^{\frac{1}{2}}$ by $x^{2} - 1$ by adding the exponents.

Tap for more steps...

Step 2.4.2.3.1

Move $x^{2} - 1$ .

$f'' (x) = - \frac{2 x^{2} - 1}{(x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{2}} x^{2}} + 0$

Step 2.4.2.3.2

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

Tap for more steps...

Step 2.4.2.3.2.1

Raise $x^{2} - 1$ to the power of $1$ .

$f'' (x) = - \frac{2 x^{2} - 1}{(x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{2}} x^{2}} + 0$

Step 2.4.2.3.2.2

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

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{1 + \frac{1}{2}} x^{2}} + 0$

Step 2.4.2.3.3

Write $1$ as a fraction with a common denominator.

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{2}{2} + \frac{1}{2}} x^{2}} + 0$

Step 2.4.2.3.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{2 + 1}{2}} x^{2}} + 0$

Step 2.4.2.3.5

Add $2$ and $1$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{3}{2}} x^{2}} + 0$

Step 2.4.2.4

Add $- \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{3}{2}} x^{2}}$ and $0$ .

$f'' (x) = - \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{3}{2}} x^{2}}$

Step 2.4.3

Reorder factors in $- \frac{2 x^{2} - 1}{{(x^{2} - 1)}^{\frac{3}{2}} x^{2}}$ .

$f'' (x) = - \frac{2 x^{2} - 1}{x^{2} {(x^{2} - 1)}^{\frac{3}{2}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{1}{x \sqrt{x^{2} - 1}} - 6 = 0$

Step 4

Simplify $\frac{1}{x \sqrt{x^{2} - 1}} - 6$ .

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{x \sqrt{x^{2} - 1^{2}}} - 6 = 0$

Step 4.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{1}{x \sqrt{(x + 1) (x - 1)}} - 6 = 0$

Step 4.1.2

Multiply $\frac{1}{x \sqrt{(x + 1) (x - 1)}}$ by $\frac{\sqrt{(x + 1) (x - 1)}}{\sqrt{(x + 1) (x - 1)}}$ .

$\frac{1}{x \sqrt{(x + 1) (x - 1)}} \cdot \frac{\sqrt{(x + 1) (x - 1)}}{\sqrt{(x + 1) (x - 1)}} - 6 = 0$

Step 4.1.3

Combine and simplify the denominator.

Tap for more steps...

Step 4.1.3.1

Multiply $\frac{1}{x \sqrt{(x + 1) (x - 1)}}$ by $\frac{\sqrt{(x + 1) (x - 1)}}{\sqrt{(x + 1) (x - 1)}}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x \sqrt{(x + 1) (x - 1)} \sqrt{(x + 1) (x - 1)}} - 6 = 0$

Step 4.1.3.2

Move $\sqrt{(x + 1) (x - 1)}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x (\sqrt{(x + 1) (x - 1)} \sqrt{(x + 1) (x - 1)})} - 6 = 0$

Step 4.1.3.3

Raise $\sqrt{(x + 1) (x - 1)}$ to the power of $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x ({\sqrt{(x + 1) (x - 1)}}^{1} \sqrt{(x + 1) (x - 1)})} - 6 = 0$

Step 4.1.3.4

Raise $\sqrt{(x + 1) (x - 1)}$ to the power of $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x ({\sqrt{(x + 1) (x - 1)}}^{1} {\sqrt{(x + 1) (x - 1)}}^{1})} - 6 = 0$

Step 4.1.3.5

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

$\frac{\sqrt{(x + 1) (x - 1)}}{x {\sqrt{(x + 1) (x - 1)}}^{1 + 1}} - 6 = 0$

Step 4.1.3.6

Add $1$ and $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {\sqrt{(x + 1) (x - 1)}}^{2}} - 6 = 0$

Step 4.1.3.7

Rewrite ${\sqrt{(x + 1) (x - 1)}}^{2}$ as $(x + 1) (x - 1)$ .

Tap for more steps...

Step 4.1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x - 1)}$ as ${((x + 1) (x - 1))}^{\frac{1}{2}}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {({((x + 1) (x - 1))}^{\frac{1}{2}})}^{2}} - 6 = 0$

Step 4.1.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{1}{2} \cdot 2}} - 6 = 0$

Step 4.1.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{2}{2}}} - 6 = 0$

Step 4.1.3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.3.7.4.1

Cancel the common factor.

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{2}{2}}} - 6 = 0$

Step 4.1.3.7.4.2

Rewrite the expression.

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{1}} - 6 = 0$

Step 4.1.3.7.5

Simplify.

$\frac{\sqrt{(x + 1) (x - 1)}}{x ((x + 1) (x - 1))} - 6 = 0$

Step 4.1.4

Simplify the denominator.

Tap for more steps...

Step 4.1.4.1

Rewrite.

$\frac{\sqrt{(x + 1) (x - 1)}}{x \sqrt{(x + 1) (x - 1)} \sqrt{(x + 1) (x - 1)}} - 6 = 0$

Step 4.1.4.2

Move $\sqrt{(x + 1) (x - 1)}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x (\sqrt{(x + 1) (x - 1)} \sqrt{(x + 1) (x - 1)})} - 6 = 0$

Step 4.1.4.3

Raise $\sqrt{(x + 1) (x - 1)}$ to the power of $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x ({\sqrt{(x + 1) (x - 1)}}^{1} \sqrt{(x + 1) (x - 1)})} - 6 = 0$

Step 4.1.4.4

Raise $\sqrt{(x + 1) (x - 1)}$ to the power of $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x ({\sqrt{(x + 1) (x - 1)}}^{1} {\sqrt{(x + 1) (x - 1)}}^{1})} - 6 = 0$

Step 4.1.4.5

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

$\frac{\sqrt{(x + 1) (x - 1)}}{x {\sqrt{(x + 1) (x - 1)}}^{1 + 1}} - 6 = 0$

Step 4.1.4.6

Add $1$ and $1$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {\sqrt{(x + 1) (x - 1)}}^{2}} - 6 = 0$

Step 4.1.4.7

Rewrite ${\sqrt{(x + 1) (x - 1)}}^{2}$ as $(x + 1) (x - 1)$ .

Tap for more steps...

Step 4.1.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x - 1)}$ as ${((x + 1) (x - 1))}^{\frac{1}{2}}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {({((x + 1) (x - 1))}^{\frac{1}{2}})}^{2}} - 6 = 0$

Step 4.1.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{1}{2} \cdot 2}} - 6 = 0$

Step 4.1.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{2}{2}}} - 6 = 0$

Step 4.1.4.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.4.7.4.1

Cancel the common factor.

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{\frac{2}{2}}} - 6 = 0$

Step 4.1.4.7.4.2

Rewrite the expression.

$\frac{\sqrt{(x + 1) (x - 1)}}{x {((x + 1) (x - 1))}^{1}} - 6 = 0$

Step 4.1.4.7.5

Simplify.

$\frac{\sqrt{(x + 1) (x - 1)}}{x ((x + 1) (x - 1))} - 6 = 0$

Step 4.1.4.8

Remove unnecessary parentheses.

$\frac{\sqrt{(x + 1) (x - 1)}}{x (x + 1) (x - 1)} - 6 = 0$

Step 4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{x (x + 1) (x - 1)}{x (x + 1) (x - 1)}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x (x + 1) (x - 1)} - 6 \cdot \frac{x (x + 1) (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.3

Simplify terms.

Tap for more steps...

Step 4.3.1

Combine $- 6$ and $\frac{x (x + 1) (x - 1)}{x (x + 1) (x - 1)}$ .

$\frac{\sqrt{(x + 1) (x - 1)}}{x (x + 1) (x - 1)} + \frac{- 6 (x (x + 1) (x - 1))}{x (x + 1) (x - 1)} = 0$

Step 4.3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{(x + 1) (x - 1)} - 6 (x (x + 1) (x - 1))}{x (x + 1) (x - 1)} = 0$

Step 4.4

Simplify the numerator.

Tap for more steps...

Step 4.4.1

Apply the distributive property.

$\frac{\sqrt{(x + 1) (x - 1)} + (- 6 x \cdot x - 6 x \cdot 1) (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.4.2.1

Move $x$ .

$\frac{\sqrt{(x + 1) (x - 1)} + (- 6 (x \cdot x) - 6 x \cdot 1) (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.2.2

Multiply $x$ by $x$ .

$\frac{\sqrt{(x + 1) (x - 1)} + (- 6 x^{2} - 6 x \cdot 1) (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.3

Multiply $- 6$ by $1$ .

$\frac{\sqrt{(x + 1) (x - 1)} + (- 6 x^{2} - 6 x) (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.4

Expand $(- 6 x^{2} - 6 x) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 4.4.4.1

Apply the distributive property.

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{2} (x - 1) - 6 x (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.4.2

Apply the distributive property.

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{2} x - 6 x^{2} \cdot - 1 - 6 x (x - 1)}{x (x + 1) (x - 1)} = 0$

Step 4.4.4.3

Apply the distributive property.

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{2} x - 6 x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5

Simplify and combine like terms.

Tap for more steps...

Step 4.4.5.1

Simplify each term.

Tap for more steps...

Step 4.4.5.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.4.5.1.1.1

Move $x$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 (x \cdot x^{2}) - 6 x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.1.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 4.4.5.1.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 (x^{1} x^{2}) - 6 x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.1.2.2

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

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{1 + 2} - 6 x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.1.3

Add $1$ and $2$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} - 6 x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.2

Multiply $- 1$ by $- 6$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 6 x^{2} - 6 x \cdot x - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.4.5.1.3.1

Move $x$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 6 x^{2} - 6 (x \cdot x) - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.3.2

Multiply $x$ by $x$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 6 x^{2} - 6 x^{2} - 6 x \cdot - 1}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.1.4

Multiply $- 1$ by $- 6$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 6 x^{2} - 6 x^{2} + 6 x}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.2

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 0 + 6 x}{x (x + 1) (x - 1)} = 0$

Step 4.4.5.3

Add $- 6 x^{3}$ and $0$ .

$\frac{\sqrt{(x + 1) (x - 1)} - 6 x^{3} + 6 x}{x (x + 1) (x - 1)} = 0$

Step 5

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.01343291$

Step 6

Evaluate the second derivative at $x = 1.01343291$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 {(1.01343291)}^{2} - 1}{{(1.01343291)}^{2} {({(1.01343291)}^{2} - 1)}^{\frac{3}{2}}}$

Step 7

Evaluate the second derivative.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raise $1.01343291$ to the power of $2$ .

$- \frac{2 \cdot 1.02704627 - 1}{{1.01343291}^{2} {({1.01343291}^{2} - 1)}^{\frac{3}{2}}}$

Step 7.1.2

Multiply $2$ by $1.02704627$ .

$- \frac{2.05409255 - 1}{{1.01343291}^{2} {({1.01343291}^{2} - 1)}^{\frac{3}{2}}}$

Step 7.1.3

Subtract $1$ from $2.05409255$ .

$- \frac{1.05409255}{{1.01343291}^{2} {({1.01343291}^{2} - 1)}^{\frac{3}{2}}}$

Step 7.2

Simplify the denominator.

Tap for more steps...

Step 7.2.1

Raise $1.01343291$ to the power of $2$ .

$- \frac{1.05409255}{1.02704627 {({1.01343291}^{2} - 1)}^{\frac{3}{2}}}$

Step 7.2.2

Raise $1.01343291$ to the power of $2$ .

$- \frac{1.05409255}{1.02704627 {(1.02704627 - 1)}^{\frac{3}{2}}}$

Step 7.2.3

Subtract $1$ from $1.02704627$ .

$- \frac{1.05409255}{1.02704627 \cdot {0.02704627}^{\frac{3}{2}}}$

Step 7.2.4

Rewrite $0.02704627$ as ${0.16445752}^{2}$ .

$- \frac{1.05409255}{1.02704627 \cdot {({0.16445752}^{2})}^{\frac{3}{2}}}$

Step 7.2.5

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1.05409255}{1.02704627 \cdot {0.16445752}^{2 (\frac{3}{2})}}$

Step 7.2.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.6.1

Cancel the common factor.

$- \frac{1.05409255}{1.02704627 \cdot {0.16445752}^{2 (\frac{3}{2})}}$

Step 7.2.6.2

Rewrite the expression.

$- \frac{1.05409255}{1.02704627 \cdot {0.16445752}^{3}}$

Step 7.2.7

Raise $0.16445752$ to the power of $3$ .

$- \frac{1.05409255}{1.02704627 \cdot 0.00444796}$

Step 7.3

Simplify the expression.

Tap for more steps...

Step 7.3.1

Multiply $1.02704627$ by $0.00444796$ .

$- \frac{1.05409255}{0.00456826}$

Step 7.3.2

Divide $1.05409255$ by $0.00456826$ .

$- 1 \cdot 230.74245167$

Step 7.3.3

Multiply $- 1$ by $230.74245167$ .

$- 230.74245167$

Step 8

$x = 1.01343291$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1.01343291$ is a local maximum

Step 9

Find the y-value when $x = 1.01343291$ .

Tap for more steps...

Step 9.1

Replace the variable $x$ with $1.01343291$ in the expression.

$f (1.01343291) = arcsec (1.01343291) - 6 \cdot 1.01343291$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Simplify each term.

Tap for more steps...

Step 9.2.1.1

Evaluate $arcsec (1.01343291)$ .

$f (1.01343291) = 0.16299847 - 6 \cdot 1.01343291$

Step 9.2.1.2

Multiply $- 6$ by $1.01343291$ .

$f (1.01343291) = 0.16299847 - 6.0805975$

Step 9.2.2

Subtract $6.0805975$ from $0.16299847$ .

$f (1.01343291) = - 5.91759902$

Step 9.2.3

The final answer is $- 5.91759902$ .

$y = - 5.91759902$

Step 10

These are the local extrema for $f (x) = arcsec (x) - 6 x$ .

$(1.01343291, - 5.91759902)$ is a local maxima

Step 11