Calculus Examples

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

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 Constant Multiple Rule.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.1.12

Simplify the expression.

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Step 1.1.12.1

Add $1$ and $0$ .

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

Step 1.1.12.2

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

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

Step 1.1.13

Simplify.

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Step 1.1.13.1

Apply the distributive property.

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

Step 1.1.13.2

Apply the distributive property.

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

Step 1.1.13.3

Combine terms.

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Step 1.1.13.3.1

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

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

Step 1.1.13.3.2

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

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

Step 1.1.13.3.3

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

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Step 1.1.13.3.3.1

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

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Step 1.1.13.3.3.1.1

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

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

Step 1.1.13.3.3.1.2

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

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

Step 1.1.13.3.3.2

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

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

Step 1.1.13.3.3.3

Combine the numerators over the common denominator.

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

Step 1.1.13.3.3.4

Subtract $1$ from $2$ .

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

Step 1.1.13.3.4

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

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

Step 1.1.13.3.5

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

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

Step 1.1.13.3.6

Cancel the common factors.

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Step 1.1.13.3.6.1

Factor $2$ out of $2$ .

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

Step 1.1.13.3.6.2

Cancel the common factor.

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

Step 1.1.13.3.6.3

Rewrite the expression.

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

Step 1.1.13.3.6.4

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

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

Step 1.1.13.3.7

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

$2 x^{\frac{1}{2}} + 4 \frac{2}{2 x^{\frac{1}{2}}} + 4 x^{\frac{1}{2}}$

Step 1.1.13.3.8

Cancel the common factor.

$2 x^{\frac{1}{2}} + 4 \frac{2}{2 x^{\frac{1}{2}}} + 4 x^{\frac{1}{2}}$

Step 1.1.13.3.9

Rewrite the expression.

$2 x^{\frac{1}{2}} + 4 \frac{1}{x^{\frac{1}{2}}} + 4 x^{\frac{1}{2}}$

Step 1.1.13.3.10

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

$2 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} + 4 x^{\frac{1}{2}}$

Step 1.1.13.3.11

Add $2 x^{\frac{1}{2}}$ and $4 x^{\frac{1}{2}}$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}}$ .

$6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} = 0$

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.

$1, x^{\frac{1}{2}}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

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

Step 2.3

Multiply each term in $6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ .

$6 x^{\frac{1}{2}} x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

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

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{\frac{1}{2}}$ .

$6 (x^{\frac{1}{2}} x^{\frac{1}{2}}) + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.1.2

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

$6 x^{\frac{1}{2} + \frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.1.3

Combine the numerators over the common denominator.

$6 x^{\frac{1 + 1}{2}} + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.1.4

Add $1$ and $1$ .

$6 x^{\frac{2}{2}} + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.1.5

Divide $2$ by $2$ .

$6 x^{1} + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.2

Simplify $6 x^{1}$ .

$6 x + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 2.3.2.1.3.1

Cancel the common factor.

$6 x + \frac{4}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.3.2

Rewrite the expression.

$6 x + 4 = 0 x^{\frac{1}{2}}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{\frac{1}{2}}$ .

$6 x + 4 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $4$ from both sides of the equation.

$6 x = - 4$

Step 2.4.2

Divide each term in $6 x = - 4$ by $6$ and simplify.

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Step 2.4.2.1

Divide each term in $6 x = - 4$ by $6$ .

$\frac{6 x}{6} = \frac{- 4}{6}$

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.

$\frac{6 x}{6} = \frac{- 4}{6}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{6}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Cancel the common factor of $- 4$ and $6$ .

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Step 2.4.2.3.1.1

Factor $2$ out of $- 4$ .

$x = \frac{2 (- 2)}{6}$

Step 2.4.2.3.1.2

Cancel the common factors.

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Step 2.4.2.3.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.4.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.4.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 2}{3}$

Step 2.4.2.3.2

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 2.5

Exclude the solutions that do not make $6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}} = 0$ true.

$No solution$

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

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.

$6 \sqrt{x^{1}} + \frac{4}{x^{\frac{1}{2}}}$

Step 4.1.2

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

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

Step 4.1.3

Anything raised to $1$ is the base itself.

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

Step 4.1.4

Anything raised to $1$ is the base itself.

$6 \sqrt{x} + \frac{4}{\sqrt{x}}$

Step 4.2

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

$\sqrt{x} = 0$

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, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

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

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Step 4.3.2.2.1.1

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

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Step 4.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 4.3.2.2.1.2

Simplify.

$x = 0^{2}$

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 $0$ .

$x = 0$

Step 4.4

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

$x < 0$

Step 4.5

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

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 5

After finding the point that makes the derivative $f' (x) = 6 x^{\frac{1}{2}} + \frac{4}{x^{\frac{1}{2}}}$ equal to $0$ or undefined, the interval to check where $f (x) = 4 (x + 2) \sqrt{x}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite ${(- 1)}^{\frac{1}{2}}$ as $\sqrt{{(- 1)}^{1}}$ .

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

Step 6.2.1.2

Evaluate the exponent.

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Simplify the denominator.

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Step 6.2.1.4.1

Rewrite ${(- 1)}^{\frac{1}{2}}$ as $\sqrt{{(- 1)}^{1}}$ .

$f' (- 1) = 6 i + \frac{4}{\sqrt{- 1}}$

Step 6.2.1.4.2

Evaluate the exponent.

$f' (- 1) = 6 i + \frac{4}{\sqrt{- 1}}$

Step 6.2.1.4.3

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

$f' (- 1) = 6 i + \frac{4}{i}$

Step 6.2.1.5

Multiply the numerator and denominator of $\frac{4}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Combine.

$f' (- 1) = 6 i + \frac{4 i}{i i}$

Step 6.2.1.6.2

Simplify the denominator.

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Step 6.2.1.6.2.1

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

$f' (- 1) = 6 i + \frac{4 i}{i i}$

Step 6.2.1.6.2.2

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

$f' (- 1) = 6 i + \frac{4 i}{i i}$

Step 6.2.1.6.2.3

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

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

Step 6.2.1.6.2.4

Add $1$ and $1$ .

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

Step 6.2.1.6.2.5

Rewrite $i^{2}$ as $- 1$ .

$f' (- 1) = 6 i + \frac{4 i}{- 1}$

Step 6.2.1.7

Move the negative one from the denominator of $\frac{4 i}{- 1}$ .

$f' (- 1) = 6 i - 1 \cdot (4 i)$

Step 6.2.1.8

Rewrite $- 1 \cdot (4 i)$ as $- (4 i)$ .

$f' (- 1) = 6 i - (4 i)$

Step 6.2.1.9

Multiply $4$ by $- 1$ .

$f' (- 1) = 6 i - 4 i$

Step 6.2.2

Subtract $4 i$ from $6 i$ .

$f' (- 1) = 2 i$

Step 6.2.3

The final answer is $2 i$ .

$2 i$

Step 6.3

At $x = - 1$ the derivative is $2 i$ . Since this contains an imaginary number, the function does not exist on $(- \infty, 0)$ .

Function is not real on $(- \infty, 0)$ since $f' (x)$ is imaginary

Step 7

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 7.1

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

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

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

One to any power is one.

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

Step 7.2.1.2

Multiply $6$ by $1$ .

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

Step 7.2.1.3

One to any power is one.

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

Step 7.2.1.4

Divide $4$ by $1$ .

$f' (1) = 6 + 4$

Step 7.2.2

Add $6$ and $4$ .

$f' (1) = 10$

Step 7.2.3

The final answer is $10$ .

$10$

Step 7.3

At $x = 1$ the derivative is $10$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 8

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

Increasing on: $(0, \infty)$

Step 9