Calculus Examples

${(x^{2} - 1)}^{\frac{2}{3}} + 5$

Step 1

Write ${(x^{2} - 1)}^{\frac{2}{3}} + 5$ as a function.

$f (x) = {(x^{2} - 1)}^{\frac{2}{3}} + 5$

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of ${(x^{2} - 1)}^{\frac{2}{3}} + 5$ with respect to $x$ is $\frac{d}{d x} [{(x^{2} - 1)}^{\frac{2}{3}}] + \frac{d}{d x} [5]$ .

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

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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 2.1.2.1.1

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

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [x^{2} - 1] + \frac{d}{d x} [5]$

Step 2.1.2.1.2

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

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} - 1] + \frac{d}{d x} [5]$

Step 2.1.2.1.3

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

Combine the numerators over the common denominator.

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

Step 2.1.2.8

Simplify the numerator.

Tap for more steps...

Step 2.1.2.8.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.8.2

Subtract $3$ from $2$ .

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

Step 2.1.2.9

Move the negative in front of the fraction.

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

Step 2.1.2.10

Add $2 x$ and $0$ .

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

Step 2.1.2.11

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

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

Step 2.1.2.12

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

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

Step 2.1.2.13

Multiply $2$ by $2$ .

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

Step 2.1.2.14

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

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

Step 2.1.2.15

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

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

Step 2.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 2.1.3.2

Add $\frac{4 x}{3 {(x^{2} - 1)}^{\frac{1}{3}}}$ and $0$ .

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

Step 2.2

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

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

Step 3

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

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

Set the numerator equal to zero.

$4 x = 0$

Step 3.3

Divide each term in $4 x = 0$ by $4$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 4

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 5

Find where the derivative is undefined.

Tap for more steps...

Step 5.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

Anything raised to $1$ is the base itself.

$\frac{4 x}{3 \sqrt[3]{x^{2} - 1}}$

Step 5.2

Set the denominator in $\frac{4 x}{3 \sqrt[3]{x^{2} - 1}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2} - 1} = 0$

Step 5.3

Solve for $x$ .

Tap for more steps...

Step 5.3.1

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

${(3 \sqrt[3]{x^{2} - 1})}^{3} = 0^{3}$

Step 5.3.2

Simplify each side of the equation.

Tap for more steps...

Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} - 1}$ as ${(x^{2} - 1)}^{\frac{1}{3}}$ .

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

Step 5.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.2.1

Simplify ${(3 {(x^{2} - 1)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 5.3.2.2.1.1

Apply the product rule to $3 {(x^{2} - 1)}^{\frac{1}{3}}$ .

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

Step 5.3.2.2.1.2

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

$27 {({(x^{2} - 1)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 5.3.2.2.1.3

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

Tap for more steps...

Step 5.3.2.2.1.3.1

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

$27 {(x^{2} - 1)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 5.3.2.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(x^{2} - 1)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 5.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x^{2} - 1)}^{1} = 0^{3}$

Step 5.3.2.2.1.4

Simplify.

$27 (x^{2} - 1) = 0^{3}$

Step 5.3.2.2.1.5

Apply the distributive property.

$27 x^{2} + 27 \cdot - 1 = 0^{3}$

Step 5.3.2.2.1.6

Multiply $27$ by $- 1$ .

$27 x^{2} - 27 = 0^{3}$

Step 5.3.2.3

Simplify the right side.

Tap for more steps...

Step 5.3.2.3.1

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

$27 x^{2} - 27 = 0$

Step 5.3.3

Solve for $x$ .

Tap for more steps...

Step 5.3.3.1

Add $27$ to both sides of the equation.

$27 x^{2} = 27$

Step 5.3.3.2

Divide each term in $27 x^{2} = 27$ by $27$ and simplify.

Tap for more steps...

Step 5.3.3.2.1

Divide each term in $27 x^{2} = 27$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{27}{27}$

Step 5.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.3.3.2.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 5.3.3.2.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{27}{27}$

Step 5.3.3.2.2.1.2

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

$x^{2} = \frac{27}{27}$

Step 5.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.2.3.1

Divide $27$ by $27$ .

$x^{2} = 1$

Step 5.3.3.3

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

$x = \pm \sqrt{1}$

Step 5.3.3.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 5.3.3.5

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

Tap for more steps...

Step 5.3.3.5.1

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

$x = 1$

Step 5.3.3.5.2

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

$x = - 1$

Step 5.3.3.5.3

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

$x = 1, - 1$

Step 5.4

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 $0$ .

$x = - 1, x = 1$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $4$ by $- 2$ .

$f' (- 2) = \frac{- 8}{3 {({(- 2)}^{2} - 1)}^{\frac{1}{3}}}$

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

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

$f' (- 2) = \frac{- 8}{3 {(4 - 1)}^{\frac{1}{3}}}$

Step 7.2.2.2

Subtract $1$ from $4$ .

$f' (- 2) = \frac{- 8}{3 \cdot 3^{\frac{1}{3}}}$

Step 7.2.3

Multiply $3$ by $3^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 7.2.3.1

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

Tap for more steps...

Step 7.2.3.1.1

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

$f' (- 2) = \frac{- 8}{3 \cdot 3^{\frac{1}{3}}}$

Step 7.2.3.1.2

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

$f' (- 2) = \frac{- 8}{3^{1 + \frac{1}{3}}}$

Step 7.2.3.2

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

$f' (- 2) = \frac{- 8}{3^{\frac{3}{3} + \frac{1}{3}}}$

Step 7.2.3.3

Combine the numerators over the common denominator.

$f' (- 2) = \frac{- 8}{3^{\frac{3 + 1}{3}}}$

Step 7.2.3.4

Add $3$ and $1$ .

$f' (- 2) = \frac{- 8}{3^{\frac{4}{3}}}$

Step 7.2.4

Move the negative in front of the fraction.

$f' (- 2) = - \frac{8}{3^{\frac{4}{3}}}$

Step 7.2.5

The final answer is $- \frac{8}{3^{\frac{4}{3}}}$ .

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

Step 7.3

At $x = - 2$ the derivative is $- \frac{8}{3^{\frac{4}{3}}}$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 1, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify the numerator.

Tap for more steps...

Step 8.2.1.1

Multiply $- 1$ by $4$ .

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

Step 8.2.1.2

Combine $- 4$ and $\frac{1}{2}$ .

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

Step 8.2.2

Simplify the denominator.

Tap for more steps...

Step 8.2.2.1

Simplify each term.

Tap for more steps...

Step 8.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.2.1.1.1

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

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

Step 8.2.2.1.1.2

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

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

Step 8.2.2.1.2

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

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

Step 8.2.2.1.3

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

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

Step 8.2.2.1.4

One to any power is one.

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

Step 8.2.2.1.5

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

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

Step 8.2.2.2

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

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

Step 8.2.2.3

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 8.2.2.4

Combine the numerators over the common denominator.

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

Step 8.2.2.5

Simplify the numerator.

Tap for more steps...

Step 8.2.2.5.1

Multiply $- 1$ by $4$ .

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

Step 8.2.2.5.2

Subtract $4$ from $1$ .

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

Step 8.2.2.6

Move the negative in front of the fraction.

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

Step 8.2.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.2.7.1

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

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

Step 8.2.2.7.2

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

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

Step 8.2.2.8

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 8.2.2.9

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

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

Step 8.2.2.10

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.2.10.1

Cancel the common factor.

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

Step 8.2.2.10.2

Rewrite the expression.

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

Step 8.2.2.11

Evaluate the exponent.

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

Step 8.2.3

Divide $- 4$ by $2$ .

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

Step 8.2.4

Simplify the denominator.

Tap for more steps...

Step 8.2.4.1

Multiply $- 1$ by $3$ .

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

Step 8.2.4.2

Combine $- 3$ and $\frac{3^{\frac{1}{3}}}{4^{\frac{1}{3}}}$ .

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

Step 8.2.5

Simplify the numerator.

Tap for more steps...

Step 8.2.5.1

Factor out negative.

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

Step 8.2.5.2

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

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

Step 8.2.5.3

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

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

Step 8.2.5.4

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

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

Step 8.2.5.5

Combine the numerators over the common denominator.

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

Step 8.2.5.6

Add $3$ and $1$ .

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

Step 8.2.6

Move the negative in front of the fraction.

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

Step 8.2.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.8

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

Tap for more steps...

Step 8.2.8.1

Multiply $- 1$ by $- 2$ .

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

Step 8.2.8.2

Combine $2$ and $\frac{4^{\frac{1}{3}}}{3^{\frac{4}{3}}}$ .

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

Step 8.2.8.3

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

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

Step 8.2.8.4

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

Tap for more steps...

Step 8.2.8.4.1

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

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

Step 8.2.8.4.2

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

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

Step 8.2.8.5

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

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

Step 8.2.8.6

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

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

Step 8.2.8.7

Combine the numerators over the common denominator.

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

Step 8.2.8.8

Add $3$ and $2$ .

$f' (- \frac{1}{2}) = \frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$

Step 8.2.9

The final answer is $\frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$ .

$\frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$

Step 8.3

At $x = - \frac{1}{2}$ the derivative is $\frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$ . Since this is positive, the function is increasing on $(- 1, 0)$ .

Increasing on $(- 1, 0)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(0, 1)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 9.2.2

Simplify the denominator.

Tap for more steps...

Step 9.2.2.1

Simplify each term.

Tap for more steps...

Step 9.2.2.1.1

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

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

Step 9.2.2.1.2

One to any power is one.

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

Step 9.2.2.1.3

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

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

Step 9.2.2.2

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

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

Step 9.2.2.3

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 9.2.2.4

Combine the numerators over the common denominator.

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

Step 9.2.2.5

Simplify the numerator.

Tap for more steps...

Step 9.2.2.5.1

Multiply $- 1$ by $4$ .

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

Step 9.2.2.5.2

Subtract $4$ from $1$ .

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

Step 9.2.2.6

Move the negative in front of the fraction.

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

Step 9.2.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 9.2.2.7.1

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

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

Step 9.2.2.7.2

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

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

Step 9.2.2.8

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 9.2.2.9

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

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

Step 9.2.2.10

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.2.2.10.1

Cancel the common factor.

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

Step 9.2.2.10.2

Rewrite the expression.

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

Step 9.2.2.11

Evaluate the exponent.

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

Step 9.2.3

Divide $4$ by $2$ .

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

Step 9.2.4

Simplify the denominator.

Tap for more steps...

Step 9.2.4.1

Multiply $- 1$ by $3$ .

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

Step 9.2.4.2

Combine $- 3$ and $\frac{3^{\frac{1}{3}}}{4^{\frac{1}{3}}}$ .

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

Step 9.2.5

Simplify the numerator.

Tap for more steps...

Step 9.2.5.1

Factor out negative.

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

Step 9.2.5.2

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

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

Step 9.2.5.3

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

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

Step 9.2.5.4

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

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

Step 9.2.5.5

Combine the numerators over the common denominator.

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

Step 9.2.5.6

Add $3$ and $1$ .

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

Step 9.2.6

Move the negative in front of the fraction.

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

Step 9.2.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.2.8

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

Tap for more steps...

Step 9.2.8.1

Multiply $- 1$ by $2$ .

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

Step 9.2.8.2

Combine $- 2$ and $\frac{4^{\frac{1}{3}}}{3^{\frac{4}{3}}}$ .

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

Step 9.2.8.3

Factor out negative.

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

Step 9.2.8.4

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

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

Step 9.2.8.5

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

Tap for more steps...

Step 9.2.8.5.1

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

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

Step 9.2.8.5.2

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

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

Step 9.2.8.6

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

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

Step 9.2.8.7

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

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

Step 9.2.8.8

Combine the numerators over the common denominator.

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

Step 9.2.8.9

Add $3$ and $2$ .

$f' (\frac{1}{2}) = \frac{- 2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$

Step 9.2.9

Move the negative in front of the fraction.

$f' (\frac{1}{2}) = - \frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$

Step 9.2.10

The final answer is $- \frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$ .

$- \frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$

Step 9.3

At $x = \frac{1}{2}$ the derivative is $- \frac{2^{\frac{5}{3}}}{3^{\frac{4}{3}}}$ . Since this is negative, the function is decreasing on $(0, 1)$ .

Decreasing on $(0, 1)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 10.1

Replace the variable $x$ with $2$ in the expression.

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

Step 10.2

Simplify the result.

Tap for more steps...

Step 10.2.1

Multiply $4$ by $2$ .

$f' (2) = \frac{8}{3 {(2^{2} - 1)}^{\frac{1}{3}}}$

Step 10.2.2

Simplify the denominator.

Tap for more steps...

Step 10.2.2.1

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

$f' (2) = \frac{8}{3 {(4 - 1)}^{\frac{1}{3}}}$

Step 10.2.2.2

Subtract $1$ from $4$ .

$f' (2) = \frac{8}{3 \cdot 3^{\frac{1}{3}}}$

Step 10.2.3

Multiply $3$ by $3^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 10.2.3.1

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

Tap for more steps...

Step 10.2.3.1.1

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

$f' (2) = \frac{8}{3 \cdot 3^{\frac{1}{3}}}$

Step 10.2.3.1.2

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

$f' (2) = \frac{8}{3^{1 + \frac{1}{3}}}$

Step 10.2.3.2

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

$f' (2) = \frac{8}{3^{\frac{3}{3} + \frac{1}{3}}}$

Step 10.2.3.3

Combine the numerators over the common denominator.

$f' (2) = \frac{8}{3^{\frac{3 + 1}{3}}}$

Step 10.2.3.4

Add $3$ and $1$ .

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

Step 10.2.4

The final answer is $\frac{8}{3^{\frac{4}{3}}}$ .

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

Step 10.3

At $x = 2$ the derivative is $\frac{8}{3^{\frac{4}{3}}}$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 11

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 1, 0), (1, \infty)$

Decreasing on: $(- \infty, - 1), (0, 1)$

Step 12