Calculus Examples

$f (x) = x^{\frac{1}{5}} (x + 6)$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $f (x) = x^{\frac{1}{5}}$ and $g (x) = x + 6$ .

$f' (x) = x^{\frac{1}{5}} \frac{d}{d x} (x + 6) + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$f' (x) = x^{\frac{1}{5}} (\frac{d}{d x} (x) + \frac{d}{d x} (6)) + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2.2

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

$f' (x) = x^{\frac{1}{5}} (1 + \frac{d}{d x} (6)) + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2.3

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

$f' (x) = x^{\frac{1}{5}} (1 + 0) + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$f' (x) = x^{\frac{1}{5}} \cdot 1 + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2.4.2

Multiply $x^{\frac{1}{5}}$ by $1$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.1.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{5}$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{1}{5} - 1})$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}})$

Step 1.1.4

Combine $- 1$ and $\frac{5}{5}$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})$

Step 1.1.5

Combine the numerators over the common denominator.

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}})$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $5$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{1 - 5}{5}})$

Step 1.1.6.2

Subtract $5$ from $1$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{\frac{- 4}{5}})$

Step 1.1.7

Move the negative in front of the fraction.

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5} x^{- \frac{4}{5}})$

Step 1.1.8

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{x^{- \frac{4}{5}}}{5})$

Step 1.1.9

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x^{\frac{1}{5}} + (x + 6) (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Apply the distributive property.

$f' (x) = x^{\frac{1}{5}} + x (\frac{1}{5 x^{\frac{4}{5}}}) + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2

Combine terms.

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Step 1.1.10.2.1

Combine $x$ and $\frac{1}{5 x^{\frac{4}{5}}}$ .

$f' (x) = x^{\frac{1}{5}} + \frac{x}{5 x^{\frac{4}{5}}} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.2

Move $x^{\frac{4}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f' (x) = x^{\frac{1}{5}} + \frac{x \cdot x^{- \frac{4}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.3

Multiply $x$ by $x^{- \frac{4}{5}}$ by adding the exponents.

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Step 1.1.10.2.3.1

Multiply $x$ by $x^{- \frac{4}{5}}$ .

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Step 1.1.10.2.3.1.1

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

$f' (x) = x^{\frac{1}{5}} + \frac{x \cdot x^{- \frac{4}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.3.1.2

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

$f' (x) = x^{\frac{1}{5}} + \frac{x^{1 - \frac{4}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.3.2

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

$f' (x) = x^{\frac{1}{5}} + \frac{x^{\frac{5}{5} - \frac{4}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.3.3

Combine the numerators over the common denominator.

$f' (x) = x^{\frac{1}{5}} + \frac{x^{\frac{5 - 4}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.3.4

Subtract $4$ from $5$ .

$f' (x) = x^{\frac{1}{5}} + \frac{x^{\frac{1}{5}}}{5} + 6 (\frac{1}{5 x^{\frac{4}{5}}})$

Step 1.1.10.2.4

Combine $6$ and $\frac{1}{5 x^{\frac{4}{5}}}$ .

$f' (x) = x^{\frac{1}{5}} + \frac{x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.1.10.2.5

To write $x^{\frac{1}{5}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f' (x) = x^{\frac{1}{5}} \cdot \frac{5}{5} + \frac{x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.1.10.2.6

Combine $x^{\frac{1}{5}}$ and $\frac{5}{5}$ .

$f' (x) = \frac{x^{\frac{1}{5}} \cdot 5}{5} + \frac{x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.1.10.2.7

Combine the numerators over the common denominator.

$f' (x) = \frac{x^{\frac{1}{5}} \cdot 5 + x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.1.10.2.8

Move $5$ to the left of $x^{\frac{1}{5}}$ .

$f' (x) = \frac{5 \cdot x^{\frac{1}{5}} + x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.1.10.2.9

Add $5 x^{\frac{1}{5}}$ and $x^{\frac{1}{5}}$ .

$f' (x) = \frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$ .

$\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5, 5 x^{\frac{4}{5}}, 1$

Step 2.2.2

Since $5, 5 x^{\frac{4}{5}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $5, 5, 1$ then find LCM for the variable part $x^{\frac{4}{5}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

Since $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $5, 5, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$5$

Step 2.2.7

The LCM of $x^{\frac{4}{5}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{4}{5}}$

Step 2.2.8

The LCM for $5, 5 x^{\frac{4}{5}}, 1$ is the numeric part $5$ multiplied by the variable part.

$5 x^{\frac{4}{5}}$

Step 2.3

Multiply each term in $\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}} = 0$ by $5 x^{\frac{4}{5}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{6 x^{\frac{1}{5}}}{5} + \frac{6}{5 x^{\frac{4}{5}}} = 0$ by $5 x^{\frac{4}{5}}$ .

$\frac{6 x^{\frac{1}{5}}}{5} (5 x^{\frac{4}{5}}) + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

$5 \frac{6 x^{\frac{1}{5}}}{5} x^{\frac{4}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.2

Cancel the common factor of $5$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$5 \frac{6 x^{\frac{1}{5}}}{5} x^{\frac{4}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.2.2

Rewrite the expression.

$6 x^{\frac{1}{5}} x^{\frac{4}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.3

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{4}{5}}$ by adding the exponents.

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Step 2.3.2.1.3.1

Move $x^{\frac{4}{5}}$ .

$6 (x^{\frac{4}{5}} x^{\frac{1}{5}}) + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.3.2

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

$6 x^{\frac{4}{5} + \frac{1}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.3.3

Combine the numerators over the common denominator.

$6 x^{\frac{4 + 1}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.3.4

Add $4$ and $1$ .

$6 x^{\frac{5}{5}} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.3.5

Divide $5$ by $5$ .

$6 x^{1} + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.4

Simplify $6 x^{1}$ .

$6 x + \frac{6}{5 x^{\frac{4}{5}}} (5 x^{\frac{4}{5}}) = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.5

Rewrite using the commutative property of multiplication.

$6 x + 5 \frac{6}{5 x^{\frac{4}{5}}} x^{\frac{4}{5}} = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.6

Cancel the common factor of $5$ .

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Step 2.3.2.1.6.1

Cancel the common factor.

$6 x + 5 \frac{6}{5 x^{\frac{4}{5}}} x^{\frac{4}{5}} = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.6.2

Rewrite the expression.

$6 x + \frac{6}{x^{\frac{4}{5}}} x^{\frac{4}{5}} = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.7

Cancel the common factor of $x^{\frac{4}{5}}$ .

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Step 2.3.2.1.7.1

Cancel the common factor.

$6 x + \frac{6}{x^{\frac{4}{5}}} x^{\frac{4}{5}} = 0 (5 x^{\frac{4}{5}})$

Step 2.3.2.1.7.2

Rewrite the expression.

$6 x + 6 = 0 (5 x^{\frac{4}{5}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (5 x^{\frac{4}{5}})$ .

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Step 2.3.3.1.1

Multiply $5$ by $0$ .

$6 x + 6 = 0 x^{\frac{4}{5}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{4}{5}}$ .

$6 x + 6 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $6$ from both sides of the equation.

$6 x = - 6$

Step 2.4.2

Divide each term in $6 x = - 6$ by $6$ and simplify.

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Step 2.4.2.1

Divide each term in $6 x = - 6$ by $6$ .

$\frac{6 x}{6} = \frac{- 6}{6}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $6$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 6}{6}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{6}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 6$ by $6$ .

$x = - 1$

Step 3

The values which make the derivative equal to $0$ are $- 1$ .

$- 1$

Step 4

Find where the derivative is undefined.

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Step 4.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

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

$\frac{6 \sqrt[5]{x^{1}}}{5} + \frac{6}{5 x^{\frac{4}{5}}}$

Step 4.1.2

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

$\frac{6 \sqrt[5]{x^{1}}}{5} + \frac{6}{5 \sqrt[5]{x^{4}}}$

Step 4.1.3

Anything raised to $1$ is the base itself.

$\frac{6 \sqrt[5]{x}}{5} + \frac{6}{5 \sqrt[5]{x^{4}}}$

Step 4.2

Set the denominator in $\frac{6}{5 \sqrt[5]{x^{4}}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{x^{4}} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(5 \sqrt[5]{x^{4}})}^{5} = 0^{5}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

${(5 x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(5 x^{\frac{4}{5}})}^{5}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $5 x^{\frac{4}{5}}$ .

$5^{5} {(x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 4.3.2.2.1.2

Raise $5$ to the power of $5$ .

$3125 {(x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 4.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{4}{5}})}^{5}$ .

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Step 4.3.2.2.1.3.1

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

$3125 x^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$3125 x^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$3125 x^{4} = 0^{5}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$3125 x^{4} = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Divide each term in $3125 x^{4} = 0$ by $3125$ and simplify.

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Step 4.3.3.1.1

Divide each term in $3125 x^{4} = 0$ by $3125$ .

$\frac{3125 x^{4}}{3125} = \frac{0}{3125}$

Step 4.3.3.1.2

Simplify the left side.

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Step 4.3.3.1.2.1

Cancel the common factor of $3125$ .

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Step 4.3.3.1.2.1.1

Cancel the common factor.

$\frac{3125 x^{4}}{3125} = \frac{0}{3125}$

Step 4.3.3.1.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{0}{3125}$

Step 4.3.3.1.3

Simplify the right side.

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Step 4.3.3.1.3.1

Divide $0$ by $3125$ .

$x^{4} = 0$

Step 4.3.3.2

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

$x = \pm \sqrt[4]{0}$

Step 4.3.3.3

Simplify $\pm \sqrt[4]{0}$ .

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Step 4.3.3.3.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 4.3.3.3.2

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

$x = \pm 0$

Step 4.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

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

Step 6

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

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Step 6.1

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

$f' (- 2) = \frac{6 {(- 2)}^{\frac{1}{5}}}{5} + \frac{6}{5 {(- 2)}^{\frac{4}{5}}}$

Step 6.2

The final answer is $\frac{6 {(- 2)}^{\frac{1}{5}}}{5} + \frac{6}{5 {(- 2)}^{\frac{4}{5}}}$ .

$\frac{6 {(- 2)}^{\frac{1}{5}}}{5} + \frac{6}{5 {(- 2)}^{\frac{4}{5}}}$

Step 6.3

At $x = - 2$ the derivative is $\frac{6 {(- 2)}^{\frac{1}{5}}}{5} + \frac{6}{5 {(- 2)}^{\frac{4}{5}}}$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

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

Step 7

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

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Step 7.1

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

$f' (- \frac{1}{2}) = \frac{6 {(- \frac{1}{2})}^{\frac{1}{5}}}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Simplify the numerator.

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Step 7.2.1.1.1

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

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Step 7.2.1.1.1.1

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

$f' (- \frac{1}{2}) = \frac{6 ({(- 1)}^{\frac{1}{5}} {(\frac{1}{2})}^{\frac{1}{5}})}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.1.2

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

$f' (- \frac{1}{2}) = \frac{6 ({(- 1)}^{\frac{1}{5}} (\frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.2

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$f' (- \frac{1}{2}) = \frac{6 ({({(- 1)}^{5})}^{\frac{1}{5}} (\frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.3

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

$f' (- \frac{1}{2}) = \frac{6 ({(- 1)}^{5 (\frac{1}{5})} (\frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.4

Cancel the common factor of $5$ .

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Step 7.2.1.1.4.1

Cancel the common factor.

$f' (- \frac{1}{2}) = \frac{6 ({(- 1)}^{5 (\frac{1}{5})} (\frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.4.2

Rewrite the expression.

$f' (- \frac{1}{2}) = \frac{6 ((- 1) (\frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.5

Evaluate the exponent.

$f' (- \frac{1}{2}) = \frac{6 (- \frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}})}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.6

One to any power is one.

$f' (- \frac{1}{2}) = \frac{6 \cdot (- 1 \frac{1}{2^{\frac{1}{5}}})}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.7

Combine exponents.

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Step 7.2.1.1.7.1

Factor out negative.

$f' (- \frac{1}{2}) = \frac{- (6 (\frac{1}{2^{\frac{1}{5}}}))}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.1.7.2

Combine $6$ and $\frac{1}{2^{\frac{1}{5}}}$ .

$f' (- \frac{1}{2}) = \frac{- \frac{6}{2^{\frac{1}{5}}}}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{1}{2}) = - \frac{6}{2^{\frac{1}{5}}} \cdot \frac{1}{5} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.3

Multiply $\frac{1}{5}$ by $\frac{6}{2^{\frac{1}{5}}}$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 {(- \frac{1}{2})}^{\frac{4}{5}}}$

Step 7.2.1.4

Simplify the denominator.

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Step 7.2.1.4.1

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

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Step 7.2.1.4.1.1

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

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({(- 1)}^{\frac{4}{5}} {(\frac{1}{2})}^{\frac{4}{5}})}$

Step 7.2.1.4.1.2

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

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({(- 1)}^{\frac{4}{5}} (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.2

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({({(- 1)}^{5})}^{\frac{4}{5}} (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.3

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

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({(- 1)}^{5 (\frac{4}{5})} (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.4

Cancel the common factor of $5$ .

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Step 7.2.1.4.4.1

Cancel the common factor.

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({(- 1)}^{5 (\frac{4}{5})} (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.4.2

Rewrite the expression.

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 ({(- 1)}^{4} (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.5

Raise $- 1$ to the power of $4$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 (1 (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}))}$

Step 7.2.1.4.6

Multiply $\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}}$ by $1$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 (\frac{1^{\frac{4}{5}}}{2^{\frac{4}{5}}})}$

Step 7.2.1.4.7

One to any power is one.

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{5 (\frac{1}{2^{\frac{4}{5}}})}$

Step 7.2.1.5

Combine $5$ and $\frac{1}{2^{\frac{4}{5}}}$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6}{\frac{5}{2^{\frac{4}{5}}}}$

Step 7.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + 6 (\frac{2^{\frac{4}{5}}}{5})$

Step 7.2.1.7

Combine $6$ and $\frac{2^{\frac{4}{5}}}{5}$ .

$f' (- \frac{1}{2}) = - \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6 \cdot 2^{\frac{4}{5}}}{5}$

Step 7.2.2

The final answer is $- \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6 \cdot 2^{\frac{4}{5}}}{5}$ .

$- \frac{6}{5 \cdot 2^{\frac{1}{5}}} + \frac{6 \cdot 2^{\frac{4}{5}}}{5}$

Step 7.3

Simplify.

$1.04466067$

Step 7.4

At $x = - \frac{1}{2}$ the derivative is $1.04466067$ . Since this is positive, the function is increasing on $(- 1, 0)$ .

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

Step 8

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

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Step 8.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{6 {(1)}^{\frac{1}{5}}}{5} + \frac{6}{5 {(1)}^{\frac{4}{5}}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

One to any power is one.

$f' (1) = \frac{6 \cdot 1}{5} + \frac{6}{5 {(1)}^{\frac{4}{5}}}$

Step 8.2.1.2

Multiply $6$ by $1$ .

$f' (1) = \frac{6}{5} + \frac{6}{5 {(1)}^{\frac{4}{5}}}$

Step 8.2.1.3

One to any power is one.

$f' (1) = \frac{6}{5} + \frac{6}{5 \cdot 1}$

Step 8.2.1.4

Multiply $5$ by $1$ .

$f' (1) = \frac{6}{5} + \frac{6}{5}$

Step 8.2.2

Combine fractions.

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Step 8.2.2.1

Combine the numerators over the common denominator.

$f' (1) = \frac{6 + 6}{5}$

Step 8.2.2.2

Add $6$ and $6$ .

$f' (1) = \frac{12}{5}$

Step 8.2.3

The final answer is $\frac{12}{5}$ .

$\frac{12}{5}$

Step 8.3

At $x = 1$ the derivative is $\frac{12}{5}$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 9

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

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

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

Step 10