Calculus Examples

$y = {(x^{2} + 1)}^{\frac{2}{3}}$

Step 1

Write $y = {(x^{2} + 1)}^{\frac{2}{3}}$ as a function.

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 2.1.1.1.1

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

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [x^{2} + 1]$

Step 2.1.1.1.2

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

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} + 1]$

Step 2.1.1.1.3

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

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

Step 2.1.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.1.1.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.1.1.4

Combine the numerators over the common denominator.

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

Step 2.1.1.5

Simplify the numerator.

Tap for more steps...

Step 2.1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 2.1.1.5.2

Subtract $3$ from $2$ .

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

Step 2.1.1.6

Combine fractions.

Tap for more steps...

Step 2.1.1.6.1

Move the negative in front of the fraction.

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

Step 2.1.1.6.2

Combine $\frac{2}{3}$ and ${(x^{2} + 1)}^{- \frac{1}{3}}$ .

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

Step 2.1.1.6.3

Move ${(x^{2} + 1)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.1.7

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

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

Step 2.1.1.8

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

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

Step 2.1.1.9

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

$\frac{2}{3 {(x^{2} + 1)}^{\frac{1}{3}}} (2 x + 0)$

Step 2.1.1.10

Combine fractions.

Tap for more steps...

Step 2.1.1.10.1

Add $2 x$ and $0$ .

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

Step 2.1.1.10.2

Combine $2$ and $\frac{2}{3 {(x^{2} + 1)}^{\frac{1}{3}}}$ .

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

Step 2.1.1.10.3

Multiply $2$ by $2$ .

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

Step 2.1.1.10.4

Combine $\frac{4}{3 {(x^{2} + 1)}^{\frac{1}{3}}}$ and $x$ .

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 x}{3 {(x^{2} + 1)}^{\frac{1}{3}}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{x}{{(x^{2} + 1)}^{\frac{1}{3}}}]$ .

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

Step 2.1.2.2

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 $f (x) = x$ and $g (x) = {(x^{2} + 1)}^{\frac{1}{3}}$ .

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

Step 2.1.2.3

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1.2.3.1

Multiply the exponents in ${({(x^{2} + 1)}^{\frac{1}{3}})}^{2}$ .

Tap for more steps...

Step 2.1.2.3.1.1

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

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

Step 2.1.2.3.1.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

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

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

Step 2.1.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 2.1.2.4.1

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

$\frac{4}{3} \cdot \frac{{(x^{2} + 1)}^{\frac{1}{3}} - x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{\frac{2}{3}}}$

Step 2.1.2.4.2

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

$\frac{4}{3} \cdot \frac{{(x^{2} + 1)}^{\frac{1}{3}} - x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{\frac{2}{3}}}$

Step 2.1.2.4.3

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

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

Step 2.1.2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.1.2.6

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.1.2.7

Combine the numerators over the common denominator.

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

Step 2.1.2.8

Simplify the numerator.

Tap for more steps...

Step 2.1.2.8.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.8.2

Subtract $3$ from $1$ .

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

Step 2.1.2.9

Combine fractions.

Tap for more steps...

Step 2.1.2.9.1

Move the negative in front of the fraction.

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

Step 2.1.2.9.2

Combine $\frac{1}{3}$ and ${(x^{2} + 1)}^{- \frac{2}{3}}$ .

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

Step 2.1.2.9.3

Move ${(x^{2} + 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.2.9.4

Combine $\frac{1}{3 {(x^{2} + 1)}^{\frac{2}{3}}}$ and $x$ .

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

Step 2.1.2.10

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

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

Step 2.1.2.11

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

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

Step 2.1.2.12

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

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

Step 2.1.2.13

Combine fractions.

Tap for more steps...

Step 2.1.2.13.1

Add $2 x$ and $0$ .

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

Step 2.1.2.13.2

Multiply $2$ by $- 1$ .

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

Step 2.1.2.13.3

Combine $- 2$ and $\frac{x}{3 {(x^{2} + 1)}^{\frac{2}{3}}}$ .

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

Step 2.1.2.13.4

Combine $\frac{- 2 x}{3 {(x^{2} + 1)}^{\frac{2}{3}}}$ and $x$ .

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

Step 2.1.2.14

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

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

Step 2.1.2.15

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

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

Step 2.1.2.16

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

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

Step 2.1.2.17

Add $1$ and $1$ .

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

Step 2.1.2.18

Move the negative in front of the fraction.

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

Step 2.1.2.19

To write ${(x^{2} + 1)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x^{2} + 1)}^{\frac{2}{3}}}{3 {(x^{2} + 1)}^{\frac{2}{3}}}$ .

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

Step 2.1.2.20

Combine ${(x^{2} + 1)}^{\frac{1}{3}}$ and $\frac{3 {(x^{2} + 1)}^{\frac{2}{3}}}{3 {(x^{2} + 1)}^{\frac{2}{3}}}$ .

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

Step 2.1.2.21

Combine the numerators over the common denominator.

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

Step 2.1.2.22

Multiply ${(x^{2} + 1)}^{\frac{1}{3}}$ by ${(x^{2} + 1)}^{\frac{2}{3}}$ by adding the exponents.

Tap for more steps...

Step 2.1.2.22.1

Move ${(x^{2} + 1)}^{\frac{2}{3}}$ .

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

Step 2.1.2.22.2

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

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

Step 2.1.2.22.3

Combine the numerators over the common denominator.

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

Step 2.1.2.22.4

Add $2$ and $1$ .

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

Step 2.1.2.22.5

Divide $3$ by $3$ .

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

Step 2.1.2.23

Simplify ${(x^{2} + 1)}^{1} \cdot 3$ .

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

Step 2.1.2.24

Move $3$ to the left of $x^{2} + 1$ .

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

Step 2.1.2.25

Rewrite $\frac{\frac{3 (x^{2} + 1) - 2 x^{2}}{3 {(x^{2} + 1)}^{\frac{2}{3}}}}{{(x^{2} + 1)}^{\frac{2}{3}}}$ as a product.

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

Step 2.1.2.26

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

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

Step 2.1.2.27

Multiply ${(x^{2} + 1)}^{\frac{2}{3}}$ by ${(x^{2} + 1)}^{\frac{2}{3}}$ by adding the exponents.

Tap for more steps...

Step 2.1.2.27.1

Move ${(x^{2} + 1)}^{\frac{2}{3}}$ .

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

Step 2.1.2.27.2

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

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

Step 2.1.2.27.3

Combine the numerators over the common denominator.

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

Step 2.1.2.27.4

Add $2$ and $2$ .

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

Step 2.1.2.28

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

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

Step 2.1.2.29

Multiply $3$ by $3$ .

$\frac{4 (3 (x^{2} + 1) - 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30

Simplify.

Tap for more steps...

Step 2.1.2.30.1

Apply the distributive property.

$\frac{4 (3 x^{2} + 3 \cdot 1 - 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.2

Apply the distributive property.

$\frac{4 (3 x^{2}) + 4 (3 \cdot 1) + 4 (- 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.3

Simplify the numerator.

Tap for more steps...

Step 2.1.2.30.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.30.3.1.1

Multiply $3$ by $4$ .

$\frac{12 x^{2} + 4 (3 \cdot 1) + 4 (- 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.3.1.2

Multiply $4 (3 \cdot 1)$ .

Tap for more steps...

Step 2.1.2.30.3.1.2.1

Multiply $3$ by $1$ .

$\frac{12 x^{2} + 4 \cdot 3 + 4 (- 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.3.1.2.2

Multiply $4$ by $3$ .

$\frac{12 x^{2} + 12 + 4 (- 2 x^{2})}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.3.1.3

Multiply $- 2$ by $4$ .

$\frac{12 x^{2} + 12 - 8 x^{2}}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.3.2

Subtract $8 x^{2}$ from $12 x^{2}$ .

$\frac{4 x^{2} + 12}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.4

Factor $4$ out of $4 x^{2} + 12$ .

Tap for more steps...

Step 2.1.2.30.4.1

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

$\frac{4 (x^{2}) + 12}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.4.2

Factor $4$ out of $12$ .

$\frac{4 x^{2} + 4 \cdot 3}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.2.30.4.3

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

$f'' (x) = \frac{4 (x^{2} + 3)}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4 (x^{2} + 3)}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$ .

$\frac{4 (x^{2} + 3)}{9 {(x^{2} + 1)}^{\frac{4}{3}}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{4 (x^{2} + 3)}{9 {(x^{2} + 1)}^{\frac{4}{3}}} = 0$ .

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{4 (x^{2} + 3)}{9 {(x^{2} + 1)}^{\frac{4}{3}}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$4 (x^{2} + 3) = 0$

Step 2.2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.2.3.1

Divide each term in $4 (x^{2} + 3) = 0$ by $4$ and simplify.

Tap for more steps...

Step 2.2.3.1.1

Divide each term in $4 (x^{2} + 3) = 0$ by $4$ .

$\frac{4 (x^{2} + 3)}{4} = \frac{0}{4}$

Step 2.2.3.1.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.1.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x^{2} + 3)}{4} = \frac{0}{4}$

Step 2.2.3.1.2.1.2

Divide $x^{2} + 3$ by $1$ .

$x^{2} + 3 = \frac{0}{4}$

Step 2.2.3.1.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1.3.1

Divide $0$ by $4$ .

$x^{2} + 3 = 0$

Step 2.2.3.2

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 2.2.3.3

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

$x = \pm \sqrt{- 3}$

Step 2.2.3.4

Simplify $\pm \sqrt{- 3}$ .

Tap for more steps...

Step 2.2.3.4.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 2.2.3.4.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 2.2.3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{3}$

Step 2.2.3.5

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

Tap for more steps...

Step 2.2.3.5.1

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

$x = i \sqrt{3}$

Step 2.2.3.5.2

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

$x = - i \sqrt{3}$

Step 2.2.3.5.3

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

$x = i \sqrt{3}, - i \sqrt{3}$

Step 3

Find the domain of $f (x) = {(x^{2} + 1)}^{\frac{2}{3}}$ .

Tap for more steps...

Step 3.1

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

$\sqrt[3]{{(x^{2} + 1)}^{2}}$

Step 3.2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \infty)$

Step 5

Substitute any number from the interval $(- \infty, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = \frac{4 ({(0)}^{2} + 3)}{9 {({(0)}^{2} + 1)}^{\frac{4}{3}}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify the numerator.

Tap for more steps...

Step 5.2.1.1

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

$f'' (0) = \frac{4 (0 + 3)}{9 {(0^{2} + 1)}^{\frac{4}{3}}}$

Step 5.2.1.2

Add $0$ and $3$ .

$f'' (0) = \frac{4 \cdot 3}{9 {(0^{2} + 1)}^{\frac{4}{3}}}$

Step 5.2.2

Simplify the denominator.

Tap for more steps...

Step 5.2.2.1

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

$f'' (0) = \frac{4 \cdot 3}{9 {(0 + 1)}^{\frac{4}{3}}}$

Step 5.2.2.2

Add $0$ and $1$ .

$f'' (0) = \frac{4 \cdot 3}{9 \cdot 1^{\frac{4}{3}}}$

Step 5.2.2.3

One to any power is one.

$f'' (0) = \frac{4 \cdot 3}{9 \cdot 1}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.3.1

Multiply $4$ by $3$ .

$f'' (0) = \frac{12}{9 \cdot 1}$

Step 5.2.3.2

Multiply $9$ by $1$ .

$f'' (0) = \frac{12}{9}$

Step 5.2.3.3

Cancel the common factor of $12$ and $9$ .

Tap for more steps...

Step 5.2.3.3.1

Factor $3$ out of $12$ .

$f'' (0) = \frac{3 (4)}{9}$

Step 5.2.3.3.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.3.2.1

Factor $3$ out of $9$ .

$f'' (0) = \frac{3 \cdot 4}{3 \cdot 3}$

Step 5.2.3.3.2.2

Cancel the common factor.

$f'' (0) = \frac{3 \cdot 4}{3 \cdot 3}$

Step 5.2.3.3.2.3

Rewrite the expression.

$f'' (0) = \frac{4}{3}$

Step 5.2.4

The final answer is $\frac{4}{3}$ .

$\frac{4}{3}$

Step 5.3

The graph is concave up on the interval $(- \infty, \infty)$ because $f'' (0)$ is positive.

The graph is concave up

Step 6