Algebra Examples

$\sqrt{x^{2} - x + 1}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

Tap for more steps...

Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Multiply $- 1$ by $1$ .

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

Step 1.13

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

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

Step 1.14

Add $2 x - 1$ and $0$ .

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

Step 1.15

Simplify.

Tap for more steps...

Step 1.15.1

Reorder the factors of $\frac{1}{2 {(x^{2} - x + 1)}^{\frac{1}{2}}} (2 x - 1)$ .

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

Step 1.15.2

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

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Multiply $2$ by $1$ .

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

Step 2.5.5

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

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

Step 2.5.6

Simplify the expression.

Tap for more steps...

Step 2.5.6.1

Add $2$ and $0$ .

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

Step 2.5.6.2

Move $2$ to the left of ${(x^{2} - x + 1)}^{\frac{1}{2}}$ .

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

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

Tap for more steps...

Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Combine fractions.

Tap for more steps...

Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

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

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

Step 2.11.3

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Multiply $- 1$ by $1$ .

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

Step 2.17

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

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

Step 2.18

Add $2 x - 1$ and $0$ .

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Add $1$ and $1$ .

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

Step 2.23

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

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

Step 2.24

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

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

Step 2.25

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

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

Step 2.26

Combine the numerators over the common denominator.

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

Step 2.27

Multiply $2$ by $2$ .

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

Step 2.28

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

Tap for more steps...

Step 2.28.1

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

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

Step 2.28.2

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

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

Step 2.28.3

Combine the numerators over the common denominator.

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

Step 2.28.4

Add $1$ and $1$ .

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

Step 2.28.5

Divide $2$ by $2$ .

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

Step 2.29

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

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

Step 2.30

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

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

Step 2.31

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

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

Step 2.32

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

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

Step 2.33

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

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

Step 2.34

Simplify the expression.

Tap for more steps...

Step 2.34.1

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

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

Step 2.34.2

Combine the numerators over the common denominator.

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

Step 2.34.3

Add $2$ and $1$ .

$f'' (x) = \frac{1}{2} \cdot \frac{4 (x^{2} - x + 1) - {(2 x - 1)}^{2}}{2 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.35

Multiply $\frac{1}{2}$ by $\frac{4 (x^{2} - x + 1) - {(2 x - 1)}^{2}}{2 {(x^{2} - x + 1)}^{\frac{3}{2}}}$ .

$f'' (x) = \frac{4 (x^{2} - x + 1) - {(2 x - 1)}^{2}}{2 (2 {(x^{2} - x + 1)}^{\frac{3}{2}})}$

Step 2.36

Multiply $2$ by $2$ .

$f'' (x) = \frac{4 (x^{2} - x + 1) - {(2 x - 1)}^{2}}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37

Simplify.

Tap for more steps...

Step 2.37.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{2} + 4 (- x) + 4 \cdot 1 - {(2 x - 1)}^{2}}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2

Simplify the numerator.

Tap for more steps...

Step 2.37.2.1

Simplify each term.

Tap for more steps...

Step 2.37.2.1.1

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 \cdot 1 - {(2 x - 1)}^{2}}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.2

Multiply $4$ by $1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - {(2 x - 1)}^{2}}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.3

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

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - ((2 x - 1) (2 x - 1))}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.4

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

Tap for more steps...

Step 2.37.2.1.4.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 x (2 x - 1) - 1 (2 x - 1))}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.4.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1))}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.4.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5

Simplify and combine like terms.

Tap for more steps...

Step 2.37.2.1.5.1

Simplify each term.

Tap for more steps...

Step 2.37.2.1.5.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 \cdot (2 x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.2

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

Tap for more steps...

Step 2.37.2.1.5.1.2.1

Move $x$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (2 \cdot (2 x^{2}) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.3

Multiply $2$ by $2$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.4

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.5

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2} - 2 x - 2 x - 1 \cdot - 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2} - 2 x - 2 x + 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.5.2

Subtract $2 x$ from $- 2 x$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2} - 4 x + 1)}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.6

Apply the distributive property.

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - (4 x^{2}) - (- 4 x) - 1 \cdot 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.7

Simplify.

Tap for more steps...

Step 2.37.2.1.7.1

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - 4 x^{2} - (- 4 x) - 1 \cdot 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.7.2

Multiply $- 4$ by $- 1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - 4 x^{2} + 4 x - 1 \cdot 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.1.7.3

Multiply $- 1$ by $1$ .

$f'' (x) = \frac{4 x^{2} - 4 x + 4 - 4 x^{2} + 4 x - 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.2

Combine the opposite terms in $4 x^{2} - 4 x + 4 - 4 x^{2} + 4 x - 1$ .

Tap for more steps...

Step 2.37.2.2.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$f'' (x) = \frac{- 4 x + 4 + 0 + 4 x - 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.2.2

Add $- 4 x + 4$ and $0$ .

$f'' (x) = \frac{- 4 x + 4 + 4 x - 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.2.3

Add $- 4 x$ and $4 x$ .

$f'' (x) = \frac{0 + 4 - 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.2.4

Add $0$ and $4$ .

$f'' (x) = \frac{4 - 1}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 2.37.2.3

Subtract $1$ from $4$ .

$f'' (x) = \frac{3}{4 {(x^{2} - x + 1)}^{\frac{3}{2}}}$

Step 3

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

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

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Simplify the numerator.

Tap for more steps...

Step 4.1.6.1

Multiply $- 1$ by $2$ .

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

Step 4.1.6.2

Subtract $2$ from $1$ .

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

Step 4.1.7

Combine fractions.

Tap for more steps...

Step 4.1.7.1

Move the negative in front of the fraction.

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.8

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

Multiply $- 1$ by $1$ .

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

Step 4.1.13

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

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

Step 4.1.14

Add $2 x - 1$ and $0$ .

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

Step 4.1.15

Simplify.

Tap for more steps...

Step 4.1.15.1

Reorder the factors of $\frac{1}{2 {(x^{2} - x + 1)}^{\frac{1}{2}}} (2 x - 1)$ .

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

Step 4.1.15.2

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

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

Step 4.2

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

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

Step 5

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

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$2 x - 1 = 0$

Step 5.3

Solve the equation for $x$ .

Tap for more steps...

Step 5.3.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 5.3.2

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

Tap for more steps...

Step 5.3.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = \frac{1}{2}$

Step 8

Evaluate the second derivative at $x = \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify the denominator.

Tap for more steps...

Step 9.1.1

Find the common denominator.

Tap for more steps...

Step 9.1.1.1

Write ${(\frac{1}{2})}^{2}$ as a fraction with denominator $1$ .

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

Step 9.1.1.2

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

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

Step 9.1.1.3

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

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

Step 9.1.1.4

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

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

Step 9.1.1.5

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.1.6

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.2

Combine the numerators over the common denominator.

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

Step 9.1.3

Simplify each term.

Tap for more steps...

Step 9.1.3.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 9.1.3.2

One to any power is one.

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

Step 9.1.3.3

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

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

Step 9.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1.3.4.1

Factor $2$ out of $4$ .

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

Step 9.1.3.4.2

Cancel the common factor.

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

Step 9.1.3.4.3

Rewrite the expression.

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

Step 9.1.4

Find the common denominator.

Tap for more steps...

Step 9.1.4.1

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

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

Step 9.1.4.2

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.4.3

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.4.4

Write $2$ as a fraction with denominator $1$ .

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

Step 9.1.4.5

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.4.6

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Step 9.1.5

Combine the numerators over the common denominator.

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

Step 9.1.6

Simplify each term.

Tap for more steps...

Step 9.1.6.1

Multiply $- 1$ by $2$ .

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

Step 9.1.6.2

Multiply $2$ by $2$ .

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

Step 9.1.7

Subtract $2$ from $1$ .

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

Step 9.1.8

Add $- 1$ and $4$ .

$\frac{3}{4 {(\frac{\frac{3}{2}}{2})}^{\frac{3}{2}}}$

Step 9.1.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.1.10

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 9.1.10.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

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

Step 9.1.10.2

Multiply $2$ by $2$ .

$\frac{3}{4 {(\frac{3}{4})}^{\frac{3}{2}}}$

Step 9.1.11

Apply the product rule to $\frac{3}{4}$ .

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{4^{\frac{3}{2}}}}$

Step 9.1.12

Simplify the denominator.

Tap for more steps...

Step 9.1.12.1

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

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{{(2^{2})}^{\frac{3}{2}}}}$

Step 9.1.12.2

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

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Step 9.1.12.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1.12.3.1

Cancel the common factor.

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Step 9.1.12.3.2

Rewrite the expression.

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{2^{3}}}$

Step 9.1.12.4

Raise $2$ to the power of $3$ .

$\frac{3}{4 \frac{3^{\frac{3}{2}}}{8}}$

Step 9.2

Simplify terms.

Tap for more steps...

Step 9.2.1

Combine $4$ and $\frac{3^{\frac{3}{2}}}{8}$ .

$\frac{3}{\frac{4 \cdot 3^{\frac{3}{2}}}{8}}$

Step 9.2.2

Reduce the expression $\frac{4 \cdot 3^{\frac{3}{2}}}{8}$ by cancelling the common factors.

Tap for more steps...

Step 9.2.2.1

Factor $4$ out of $4 \cdot 3^{\frac{3}{2}}$ .

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

Step 9.2.2.2

Factor $4$ out of $8$ .

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

Step 9.2.2.3

Cancel the common factor.

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

Step 9.2.2.4

Rewrite the expression.

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

Step 9.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.4

Multiply $3 \frac{2}{3^{\frac{3}{2}}}$ .

Tap for more steps...

Step 9.4.1

Combine $3$ and $\frac{2}{3^{\frac{3}{2}}}$ .

$\frac{3 \cdot 2}{3^{\frac{3}{2}}}$

Step 9.4.2

Multiply $3$ by $2$ .

$\frac{6}{3^{\frac{3}{2}}}$

Step 10

$x = \frac{1}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{2}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{2}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 11.2.2

One to any power is one.

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

Step 11.2.3

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

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

Step 11.2.4

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

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

Step 11.2.5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 11.2.5.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

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

Step 11.2.5.2

Multiply $2$ by $2$ .

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

Step 11.2.6

Combine the numerators over the common denominator.

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

Step 11.2.7

Subtract $2$ from $1$ .

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

Step 11.2.8

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

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

Step 11.2.9

Combine the numerators over the common denominator.

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

Step 11.2.10

Add $- 1$ and $4$ .

$f (\frac{1}{2}) = \sqrt{\frac{3}{4}}$

Step 11.2.11

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$f (\frac{1}{2}) = \frac{\sqrt{3}}{\sqrt{4}}$

Step 11.2.12

Simplify the denominator.

Tap for more steps...

Step 11.2.12.1

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

$f (\frac{1}{2}) = \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 11.2.12.2

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

$f (\frac{1}{2}) = \frac{\sqrt{3}}{2}$

Step 11.2.13

The final answer is $\frac{\sqrt{3}}{2}$ .

$y = \frac{\sqrt{3}}{2}$

Step 12

These are the local extrema for $f (x) = \sqrt{x^{2} - x + 1}$ .

$(\frac{1}{2}, \frac{\sqrt{3}}{2})$ is a local minima

Step 13