Calculus Examples

$x \sqrt{4 - x^{2}}$

Step 1

Write $x \sqrt{4 - x^{2}}$ as a function.

$f (x) = x \sqrt{4 - x^{2}}$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(4 - x^{2})}^{\frac{1}{2}}]$

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 $f (x) = x$ and $g (x) = {(4 - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x^{2}$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

Tap for more steps...

Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $1$ .

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

Step 2.8

Combine fractions.

Tap for more steps...

Step 2.8.1

Move the negative in front of the fraction.

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

Step 2.8.2

Combine $\frac{1}{2}$ and ${(4 - x^{2})}^{- \frac{1}{2}}$ .

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

Step 2.8.3

Move ${(4 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8.4

Combine $\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 2.9

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

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

Step 2.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 2.12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.13

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

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

Step 2.14

Combine fractions.

Tap for more steps...

Step 2.14.1

Multiply $2$ by $- 1$ .

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

Step 2.14.2

Combine $- 2$ and $\frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 2.14.3

Combine $\frac{- 2 x}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

Add $1$ and $1$ .

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

Step 2.19

Factor $2$ out of $- 2 x^{2}$ .

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

Step 2.20

Cancel the common factors.

Tap for more steps...

Step 2.20.1

Factor $2$ out of $2 {(4 - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.20.2

Cancel the common factor.

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

Step 2.20.3

Rewrite the expression.

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

Step 2.21

Move the negative in front of the fraction.

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

Step 2.22

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

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \cdot 1$

Step 2.23

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}}$

Step 2.24

To write ${(4 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + \frac{{(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.25

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.26

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by ${(4 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 2.26.1

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

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.26.2

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1 + 1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.26.3

Add $1$ and $1$ .

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{2}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.26.4

Divide $2$ by $2$ .

$\frac{- x^{2} + {(4 - x^{2})}^{1}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.27

Simplify ${(4 - x^{2})}^{1}$ .

$\frac{- x^{2} + 4 - x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.28

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

$\frac{- 2 x^{2} + 4}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.29

Reorder terms.

$\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = - 2 x^{2} + 4$ and $g (x) = {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}}$

Step 3.2

Multiply the exponents in ${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2.2

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.3

Simplify.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4

Differentiate.

Tap for more steps...

Step 3.4.1

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (\frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 2 \frac{d}{d x} x^{2} + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4.3

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 2 (2 x) + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4.4

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x + 0) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 3.4.6

Add $- 4 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = - x^{2} + 4$ .

Tap for more steps...

Step 3.5.1

To apply the Chain Rule, set $u$ as $- x^{2} + 4$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.5.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.5.3

Replace all occurrences of $u$ with $- x^{2} + 4$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.6

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.7

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.9

Simplify the numerator.

Tap for more steps...

Step 3.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.9.2

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{- 1}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.10

Combine fractions.

Tap for more steps...

Step 3.10.1

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{- \frac{1}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.10.2

Combine $\frac{1}{2}$ and ${(- x^{2} + 4)}^{- \frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{{(- x^{2} + 4)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.10.3

Move ${(- x^{2} + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 3.11

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 3.12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2} + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 3.13

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- (2 x) + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 3.14

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 3.15

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x + 0))}{- x^{2} + 4}$

Step 3.16

Simplify terms.

Tap for more steps...

Step 3.16.1

Add $- 2 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x))}{- x^{2} + 4}$

Step 3.16.2

Combine $- 2$ and $\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{- 2}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot x)}{- x^{2} + 4}$

Step 3.16.3

Combine $\frac{- 2}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{- 2 x}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.16.4

Factor $2$ out of $- 2 x$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.17

Cancel the common factors.

Tap for more steps...

Step 3.17.1

Factor $2$ out of $2 {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 ({(- x^{2} + 4)}^{\frac{1}{2}})}}{- x^{2} + 4}$

Step 3.17.2

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.17.3

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{- x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.18

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (- \frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 3.19

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) + 1 (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 3.20

Multiply $- 2 x^{2} + 4$ by $1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) + (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 3.21

Simplify.

Tap for more steps...

Step 3.21.1

Simplify the numerator.

Tap for more steps...

Step 3.21.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 3.21.1.2

Multiply $- 2 x^{2} + 4$ by $\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} + 4) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.3

Factor $2$ out of $- 2 x^{2} + 4$ .

Tap for more steps...

Step 3.21.1.3.1

Factor $2$ out of $- 2 x^{2}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 4) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.3.2

Factor $2$ out of $4$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 2 (2)) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.3.3

Factor $2$ out of $2 (- x^{2}) + 2 (2)$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.4

To write $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x \cdot \frac{{(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.5

Combine $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x$ and $\frac{{(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7

Rewrite $\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ in a factored form.

Tap for more steps...

Step 3.21.1.7.1

Factor $2 x$ out of $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x$ .

Tap for more steps...

Step 3.21.1.7.1.1

Factor $2 x$ out of $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.1.2

Factor $2 x$ out of $2 (- x^{2} + 2) x$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 x (- x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.1.3

Factor $2 x$ out of $2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 x (- x^{2} + 2)$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2

Combine exponents.

Tap for more steps...

Step 3.21.1.7.2.1

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by ${(- x^{2} + 4)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 3.21.1.7.2.1.1

Move ${(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 ({(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2.1.2

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

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2} + \frac{1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2.1.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1 + 1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2.1.4

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{2}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2.1.5

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 4) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.7.2.2

Simplify $- 2 {(- x^{2} + 4)}^{1}$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 4) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.8

Simplify the numerator.

Tap for more steps...

Step 3.21.1.8.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2}) - 2 \cdot 4 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.8.2

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 2 \cdot 4 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.8.3

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 8 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.8.4

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

$f'' (x) = \frac{\frac{2 x (x^{2} - 8 + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.1.8.5

Add $- 8$ and $2$ .

$f'' (x) = \frac{\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 3.21.2

Combine terms.

Tap for more steps...

Step 3.21.2.1

Rewrite $\frac{\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$ as a product.

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}} \cdot \frac{1}{- x^{2} + 4}$

Step 3.21.2.2

Multiply $\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ by $\frac{1}{- x^{2} + 4}$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}} (- x^{2} + 4)}$

Step 3.21.2.3

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by $- x^{2} + 4$ by adding the exponents.

Tap for more steps...

Step 3.21.2.3.1

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by $- x^{2} + 4$ .

Tap for more steps...

Step 3.21.2.3.1.1

Raise $- x^{2} + 4$ to the power of $1$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}} (- x^{2} + 4)}$

Step 3.21.2.3.1.2

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

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2} + 1}}$

Step 3.21.2.3.2

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

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 3.21.2.3.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1 + 2}{2}}}$

Step 3.21.2.3.4

Add $1$ and $2$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}}$

Step 4

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

$\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

Tap for more steps...

Step 5.1

Find the first derivative.

Tap for more steps...

Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(4 - x^{2})}^{\frac{1}{2}}]$

Step 5.1.2

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

Step 5.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x^{2}$ .

Tap for more steps...

Step 5.1.3.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

Combine the numerators over the common denominator.

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

Step 5.1.7

Simplify the numerator.

Tap for more steps...

Step 5.1.7.1

Multiply $- 1$ by $2$ .

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

Step 5.1.7.2

Subtract $2$ from $1$ .

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

Step 5.1.8

Combine fractions.

Tap for more steps...

Step 5.1.8.1

Move the negative in front of the fraction.

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

Step 5.1.8.2

Combine $\frac{1}{2}$ and ${(4 - x^{2})}^{- \frac{1}{2}}$ .

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

Step 5.1.8.3

Move ${(4 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.1.8.4

Combine $\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 5.1.9

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

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

Step 5.1.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 5.1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 5.1.12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 5.1.13

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

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

Step 5.1.14

Combine fractions.

Tap for more steps...

Step 5.1.14.1

Multiply $2$ by $- 1$ .

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

Step 5.1.14.2

Combine $- 2$ and $\frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 5.1.14.3

Combine $\frac{- 2 x}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 5.1.15

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

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

Step 5.1.16

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

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

Step 5.1.17

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

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

Step 5.1.18

Add $1$ and $1$ .

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

Step 5.1.19

Factor $2$ out of $- 2 x^{2}$ .

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

Step 5.1.20

Cancel the common factors.

Tap for more steps...

Step 5.1.20.1

Factor $2$ out of $2 {(4 - x^{2})}^{\frac{1}{2}}$ .

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

Step 5.1.20.2

Cancel the common factor.

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

Step 5.1.20.3

Rewrite the expression.

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

Step 5.1.21

Move the negative in front of the fraction.

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

Step 5.1.22

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

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \cdot 1$

Step 5.1.23

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}}$

Step 5.1.24

To write ${(4 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + \frac{{(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.25

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.26

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by ${(4 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 5.1.26.1

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

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.26.2

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{1 + 1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.26.3

Add $1$ and $1$ .

$\frac{- x^{2} + {(4 - x^{2})}^{\frac{2}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.26.4

Divide $2$ by $2$ .

$\frac{- x^{2} + {(4 - x^{2})}^{1}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.27

Simplify ${(4 - x^{2})}^{1}$ .

$\frac{- x^{2} + 4 - x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.28

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

$\frac{- 2 x^{2} + 4}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.29

Reorder terms.

$f' (x) = \frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}} = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

$\frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}} = 0$

Step 6.2

Set the numerator equal to zero.

$- 2 x^{2} + 4 = 0$

Step 6.3

Solve the equation for $x$ .

Tap for more steps...

Step 6.3.1

Subtract $4$ from both sides of the equation.

$- 2 x^{2} = - 4$

Step 6.3.2

Divide each term in $- 2 x^{2} = - 4$ by $- 2$ and simplify.

Tap for more steps...

Step 6.3.2.1

Divide each term in $- 2 x^{2} = - 4$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 4}{- 2}$

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 4}{- 2}$

Step 6.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 2}$

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

Divide $- 4$ by $- 2$ .

$x^{2} = 2$

Step 6.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 6.3.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 6.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 6.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 7

Find the values where the derivative is undefined.

Tap for more steps...

Step 7.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 7.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{- 2 x^{2} + 4}{\sqrt{{(- x^{2} + 4)}^{1}}}$

Step 7.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 4}{\sqrt{- x^{2} + 4}}$

Step 7.2

Set the denominator in $\frac{- 2 x^{2} + 4}{\sqrt{- x^{2} + 4}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 4} = 0$

Step 7.3

Solve for $x$ .

Tap for more steps...

Step 7.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- x^{2} + 4}}^{2} = 0^{2}$

Step 7.3.2

Simplify each side of the equation.

Tap for more steps...

Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x^{2} + 4}$ as ${(- x^{2} + 4)}^{\frac{1}{2}}$ .

${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 7.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.2.1

Simplify ${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 7.3.2.2.1.1

Multiply the exponents in ${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 7.3.2.2.1.1.1

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

${(- x^{2} + 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.2.2.1.1.2.1

Cancel the common factor.

${(- x^{2} + 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.1.2.2

Rewrite the expression.

${(- x^{2} + 4)}^{1} = 0^{2}$

Step 7.3.2.2.1.2

Simplify.

$- x^{2} + 4 = 0^{2}$

Step 7.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$- x^{2} + 4 = 0$

Step 7.3.3

Solve for $x$ .

Tap for more steps...

Step 7.3.3.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 7.3.3.2

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 7.3.3.2.1

Divide each term in $- x^{2} = - 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Step 7.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Step 7.3.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 1}$

Step 7.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.3.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 7.3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 7.3.3.4

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 7.3.3.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 7.3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 7.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 7.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 7.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 7.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 7.4

Set the radicand in $\sqrt{- x^{2} + 4}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 4 < 0$

Step 7.5

Solve for $x$ .

Tap for more steps...

Step 7.5.1

Subtract $4$ from both sides of the inequality.

$- x^{2} < - 4$

Step 7.5.2

Divide each term in $- x^{2} < - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 7.5.2.1

Divide each term in $- x^{2} < - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 4}{- 1}$

Step 7.5.2.2

Simplify the left side.

Tap for more steps...

Step 7.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 4}{- 1}$

Step 7.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 4}{- 1}$

Step 7.5.2.3

Simplify the right side.

Tap for more steps...

Step 7.5.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} > 4$

Step 7.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{4}$

Step 7.5.4

Simplify the equation.

Tap for more steps...

Step 7.5.4.1

Simplify the left side.

Tap for more steps...

Step 7.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{4}$

Step 7.5.4.2

Simplify the right side.

Tap for more steps...

Step 7.5.4.2.1

Simplify $\sqrt{4}$ .

Tap for more steps...

Step 7.5.4.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | > \sqrt{2^{2}}$

Step 7.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 2 |$

Step 7.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | > 2$

Step 7.5.5

Write $| x | > 2$ as a piecewise.

Tap for more steps...

Step 7.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 7.5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 2$

Step 7.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 7.5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 2$

Step 7.5.5.5

Write as a piecewise.

${\begin{cases} x > 2 & x \geq 0 \\ - x > 2 & x < 0 \end{cases}$

Step 7.5.6

Find the intersection of $x > 2$ and $x \geq 0$ .

$x > 2$

Step 7.5.7

Divide each term in $- x > 2$ by $- 1$ and simplify.

Tap for more steps...

Step 7.5.7.1

Divide each term in $- x > 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{2}{- 1}$

Step 7.5.7.2

Simplify the left side.

Tap for more steps...

Step 7.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{2}{- 1}$

Step 7.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{2}{- 1}$

Step 7.5.7.3

Simplify the right side.

Tap for more steps...

Step 7.5.7.3.1

Divide $2$ by $- 1$ .

$x < - 2$

Step 7.5.8

Find the union of the solutions.

$x < - 2$ or $x > 2$

Step 7.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

Step 8

Critical points to evaluate.

$x = \sqrt{2}, - \sqrt{2}, - 2, 2$

Step 9

Evaluate the second derivative at $x = \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (\sqrt{2}) ({(\sqrt{2})}^{2} - 6)}{{(- {(\sqrt{2})}^{2} + 4)}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 10.1.1.1

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

$\frac{2 \sqrt{2} ({(2^{\frac{1}{2}})}^{2} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.2

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

$\frac{2 \sqrt{2} (2^{\frac{1}{2} \cdot 2} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.3

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

$\frac{2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1.1.4.1

Cancel the common factor.

$\frac{2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.4.2

Rewrite the expression.

$\frac{2 \sqrt{2} (2^{1} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.5

Evaluate the exponent.

$\frac{2 \sqrt{2} (2 - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.2

Subtract $6$ from $2$ .

$\frac{2 \sqrt{2} \cdot - 4}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.1.3

Multiply $- 4$ by $2$ .

$\frac{- 8 \sqrt{2}}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Simplify each term.

Tap for more steps...

Step 10.2.1.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 10.2.1.1.1

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

$\frac{- 8 \sqrt{2}}{{(- {(2^{\frac{1}{2}})}^{2} + 4)}^{\frac{3}{2}}}$

Step 10.2.1.1.2

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

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{1}{2} \cdot 2} + 4)}^{\frac{3}{2}}}$

Step 10.2.1.1.3

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

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 10.2.1.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.1.1.4.1

Cancel the common factor.

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 10.2.1.1.4.2

Rewrite the expression.

$\frac{- 8 \sqrt{2}}{{(- 2^{1} + 4)}^{\frac{3}{2}}}$

Step 10.2.1.1.5

Evaluate the exponent.

$\frac{- 8 \sqrt{2}}{{(- 1 \cdot 2 + 4)}^{\frac{3}{2}}}$

Step 10.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- 8 \sqrt{2}}{{(- 2 + 4)}^{\frac{3}{2}}}$

Step 10.2.2

Add $- 2$ and $4$ .

$\frac{- 8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 11

$x = \sqrt{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \sqrt{2}$ is a local maximum

Step 12

Find the y-value when $x = \sqrt{2}$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

$f (\sqrt{2}) = (\sqrt{2}) \sqrt{4 - {(\sqrt{2})}^{2}}$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Combine using the product rule for radicals.

$f (\sqrt{2}) = \sqrt{2 (4 - {(\sqrt{2})}^{2})}$

Step 12.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 12.2.2.1

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

$f (\sqrt{2}) = \sqrt{2 (4 - {(2^{\frac{1}{2}})}^{2})}$

Step 12.2.2.2

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

$f (\sqrt{2}) = \sqrt{2 (4 - 2^{\frac{1}{2} \cdot 2})}$

Step 12.2.2.3

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

$f (\sqrt{2}) = \sqrt{2 (4 - 2^{\frac{2}{2}})}$

Step 12.2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.2.2.4.1

Cancel the common factor.

$f (\sqrt{2}) = \sqrt{2 (4 - 2^{\frac{2}{2}})}$

Step 12.2.2.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 12.2.2.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2 (4 - 1 \cdot 2)}$

Step 12.2.3

Simplify the expression.

Tap for more steps...

Step 12.2.3.1

Multiply $- 1$ by $2$ .

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 12.2.3.2

Subtract $2$ from $4$ .

$f (\sqrt{2}) = \sqrt{2 \cdot 2}$

Step 12.2.3.3

Multiply $2$ by $2$ .

$f (\sqrt{2}) = \sqrt{4}$

Step 12.2.3.4

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{2}) = \sqrt{2^{2}}$

Step 12.2.4

Pull terms out from under the radical, assuming positive real numbers.

$f (\sqrt{2}) = 2$

Step 12.2.5

The final answer is $2$ .

$y = 2$

Step 13

Evaluate the second derivative at $x = - \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (- \sqrt{2}) ({(- \sqrt{2})}^{2} - 6)}{{(- {(- \sqrt{2})}^{2} + 4)}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Multiply $- 1$ by $2$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- {(- \sqrt{2})}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2

Simplify the denominator.

Tap for more steps...

Step 14.2.1

Simplify each term.

Tap for more steps...

Step 14.2.1.1

Apply the product rule to $- \sqrt{2}$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- ({(- 1)}^{2} {\sqrt{2}}^{2}) + 4)}^{\frac{3}{2}}}$

Step 14.2.1.2

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

Tap for more steps...

Step 14.2.1.2.1

Move ${(- 1)}^{2}$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{({(- 1)}^{2} \cdot - 1 {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.2.2

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

Tap for more steps...

Step 14.2.1.2.2.1

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

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{({(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.2.2.2

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

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{({(- 1)}^{2 + 1} {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.2.3

Add $2$ and $1$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{({(- 1)}^{3} {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.3

Raise $- 1$ to the power of $3$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 14.2.1.4.1

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

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- {(2^{\frac{1}{2}})}^{2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4.2

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

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 2^{\frac{1}{2} \cdot 2} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4.3

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

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 14.2.1.4.4.1

Cancel the common factor.

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4.4.2

Rewrite the expression.

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 2^{1} + 4)}^{\frac{3}{2}}}$

Step 14.2.1.4.5

Evaluate the exponent.

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 1 \cdot 2 + 4)}^{\frac{3}{2}}}$

Step 14.2.1.5

Multiply $- 1$ by $2$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{{(- 2 + 4)}^{\frac{3}{2}}}$

Step 14.2.2

Add $- 2$ and $4$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3

Simplify the numerator.

Tap for more steps...

Step 14.3.1

Apply the product rule to $- \sqrt{2}$ .

$\frac{- 2 \sqrt{2} ({(- 1)}^{2} {\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.2

Raise $- 1$ to the power of $2$ .

$\frac{- 2 \sqrt{2} (1 {\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$\frac{- 2 \sqrt{2} ({\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 14.3.4.1

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

$\frac{- 2 \sqrt{2} ({(2^{\frac{1}{2}})}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4.2

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

$\frac{- 2 \sqrt{2} (2^{\frac{1}{2} \cdot 2} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4.3

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

$\frac{- 2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 14.3.4.4.1

Cancel the common factor.

$\frac{- 2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4.4.2

Rewrite the expression.

$\frac{- 2 \sqrt{2} (2^{1} - 6)}{2^{\frac{3}{2}}}$

Step 14.3.4.5

Evaluate the exponent.

$\frac{- 2 \sqrt{2} (2 - 6)}{2^{\frac{3}{2}}}$

Step 14.3.5

Subtract $6$ from $2$ .

$\frac{- 2 \sqrt{2} \cdot - 4}{2^{\frac{3}{2}}}$

Step 14.3.6

Multiply $- 4$ by $- 2$ .

$\frac{8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 15

$x = - \sqrt{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \sqrt{2}$ is a local minimum

Step 16

Find the y-value when $x = - \sqrt{2}$ .

Tap for more steps...

Step 16.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

$f (- \sqrt{2}) = (- \sqrt{2}) \sqrt{4 - {(- \sqrt{2})}^{2}}$

Step 16.2

Simplify the result.

Tap for more steps...

Step 16.2.1

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - ({(- 1)}^{2} {\sqrt{2}}^{2})}$

Step 16.2.2

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

Tap for more steps...

Step 16.2.2.1

Move ${(- 1)}^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{2} \cdot (- 1 {\sqrt{2}}^{2})}$

Step 16.2.2.2

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

Tap for more steps...

Step 16.2.2.2.1

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{2}}^{2})}$

Step 16.2.2.2.2

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{2 + 1} {\sqrt{2}}^{2}}$

Step 16.2.2.3

Add $2$ and $1$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{3} {\sqrt{2}}^{2}}$

Step 16.2.3

Raise $- 1$ to the power of $3$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - {\sqrt{2}}^{2}}$

Step 16.2.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 16.2.4.1

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - {(2^{\frac{1}{2}})}^{2}}$

Step 16.2.4.2

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2^{\frac{1}{2} \cdot 2}}$

Step 16.2.4.3

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2^{\frac{2}{2}}}$

Step 16.2.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.2.4.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2^{\frac{2}{2}}}$

Step 16.2.4.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2}$

Step 16.2.4.5

Evaluate the exponent.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 1 \cdot 2}$

Step 16.2.5

Simplify the expression.

Tap for more steps...

Step 16.2.5.1

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2}$

Step 16.2.5.2

Subtract $2$ from $4$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2}$

Step 16.2.6

Multiply $- \sqrt{2} \sqrt{2}$ .

Tap for more steps...

Step 16.2.6.1

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - (\sqrt{2} \sqrt{2})$

Step 16.2.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - (\sqrt{2} \sqrt{2})$

Step 16.2.6.3

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

$f (- \sqrt{2}) = - {\sqrt{2}}^{1 + 1}$

Step 16.2.6.4

Add $1$ and $1$ .

$f (- \sqrt{2}) = - {\sqrt{2}}^{2}$

Step 16.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 16.2.7.1

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

$f (- \sqrt{2}) = - {(2^{\frac{1}{2}})}^{2}$

Step 16.2.7.2

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

$f (- \sqrt{2}) = - 2^{\frac{1}{2} \cdot 2}$

Step 16.2.7.3

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

$f (- \sqrt{2}) = - 2^{\frac{2}{2}}$

Step 16.2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.2.7.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - 2^{\frac{2}{2}}$

Step 16.2.7.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - 2$

Step 16.2.7.5

Evaluate the exponent.

$f (- \sqrt{2}) = - 1 \cdot 2$

Step 16.2.8

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - 2$

Step 16.2.9

The final answer is $- 2$ .

$y = - 2$

Step 17

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{{(- {(- 2)}^{2} + 4)}^{\frac{3}{2}}}$

Step 18

Evaluate the second derivative.

Tap for more steps...

Step 18.1

Simplify each term.

Tap for more steps...

Step 18.1.1

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

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{{(- 1 \cdot 4 + 4)}^{\frac{3}{2}}}$

Step 18.1.2

Multiply $- 1$ by $4$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{{(- 4 + 4)}^{\frac{3}{2}}}$

Step 18.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 18.2.1

Add $- 4$ and $4$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{\frac{3}{2}}}$

Step 18.2.2

Simplify the expression.

Tap for more steps...

Step 18.2.2.1

Rewrite $0$ as $0^{2}$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{{(0^{2})}^{\frac{3}{2}}}$

Step 18.2.2.2

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

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{2 (\frac{3}{2})}}$

Step 18.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.2.3.1

Cancel the common factor.

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{2 (\frac{3}{2})}}$

Step 18.2.3.2

Rewrite the expression.

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{3}}$

Step 18.2.4

Raising $0$ to any positive power yields $0$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0}$

Step 18.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0}$

Step 18.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 19

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 20