Calculus Examples

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

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

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

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

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

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

Step 1.1.3

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 $g (x) = x + 2$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $x + 2$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Replace all occurrences of $u$ with $x + 2$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

Simplify the expression.

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Step 1.1.12.1

Add $1$ and $0$ .

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

Step 1.1.12.2

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

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

Step 1.1.13

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

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

Step 1.1.14

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

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

Step 1.1.15

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

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

Step 1.1.16

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

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

Step 1.1.17

Combine the numerators over the common denominator.

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

Step 1.1.18

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

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Step 1.1.18.1

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

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

Step 1.1.18.2

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

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

Step 1.1.18.3

Combine the numerators over the common denominator.

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

Step 1.1.18.4

Add $1$ and $1$ .

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

Step 1.1.18.5

Divide $2$ by $2$ .

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

Step 1.1.19

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

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

Step 1.1.20

Move $2$ to the left of $x + 2$ .

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

Step 1.1.21

Simplify.

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Step 1.1.21.1

Apply the distributive property.

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

Step 1.1.21.2

Simplify the numerator.

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Step 1.1.21.2.1

Multiply $2$ by $2$ .

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

Step 1.1.21.2.2

Add $x$ and $2 x$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$3 x + 4 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 2.3.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 2.3.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 3

The values which make the derivative equal to $0$ are $- \frac{4}{3}$ .

$- \frac{4}{3}$

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.

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

Step 4.1.2

Anything raised to $1$ is the base itself.

$\frac{3 x + 4}{2 \sqrt{x + 2}}$

Step 4.2

Set the denominator in $\frac{3 x + 4}{2 \sqrt{x + 2}}$ equal to $0$ to find where the expression is undefined.

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

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.

${(2 \sqrt{x + 2})}^{2} = 0^{2}$

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

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

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

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Step 4.3.2.2.1.1

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

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

Step 4.3.2.2.1.2

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

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

Step 4.3.2.2.1.3

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

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Step 4.3.2.2.1.3.1

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

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

Step 4.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 4.3.2.2.1.4

Simplify.

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

Step 4.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 2 = 0^{2}$

Step 4.3.2.2.1.6

Multiply $4$ by $2$ .

$4 x + 8 = 0^{2}$

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 $0$ .

$4 x + 8 = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Subtract $8$ from both sides of the equation.

$4 x = - 8$

Step 4.3.3.2

Divide each term in $4 x = - 8$ by $4$ and simplify.

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Step 4.3.3.2.1

Divide each term in $4 x = - 8$ by $4$ .

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 4.3.3.2.2

Simplify the left side.

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Step 4.3.3.2.2.1

Cancel the common factor of $4$ .

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Step 4.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 4.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{4}$

Step 4.3.3.2.3

Simplify the right side.

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Step 4.3.3.2.3.1

Divide $- 8$ by $4$ .

$x = - 2$

Step 4.4

Set the radicand in $\sqrt{x + 2}$ less than $0$ to find where the expression is undefined.

$x + 2 < 0$

Step 4.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 4.6

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 $0$ .

$x \leq - 2$

$(- \infty, - 2]$

$x \leq - 2$

$(- \infty, - 2]$

Step 5

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

$(- \infty, - 2) \cup (- 2, - \frac{4}{3}) \cup (- \frac{4}{3}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $3$ by $- 3$ .

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

Step 6.2.1.2

Add $- 9$ and $4$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 3$ and $2$ .

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Evaluate the exponent.

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

Step 6.2.2.4

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

$f' (- 3) = \frac{- 5}{2 i}$

Step 6.2.3

Multiply the numerator and denominator of $\frac{- 5}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$f' (- 3) = \frac{- 5}{2 i} \cdot \frac{i}{i}$

Step 6.2.4

Multiply.

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Step 6.2.4.1

Combine.

$f' (- 3) = \frac{- 5 i}{2 i i}$

Step 6.2.4.2

Simplify the denominator.

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Step 6.2.4.2.1

Add parentheses.

$f' (- 3) = \frac{- 5 i}{2 (i i)}$

Step 6.2.4.2.2

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

$f' (- 3) = \frac{- 5 i}{2 (i i)}$

Step 6.2.4.2.3

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

$f' (- 3) = \frac{- 5 i}{2 (i i)}$

Step 6.2.4.2.4

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

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

Step 6.2.4.2.5

Add $1$ and $1$ .

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

Step 6.2.4.2.6

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

$f' (- 3) = \frac{- 5 i}{2 \cdot - 1}$

Step 6.2.5

Multiply $2$ by $- 1$ .

$f' (- 3) = \frac{- 5 i}{- 2}$

Step 6.2.6

Dividing two negative values results in a positive value.

$f' (- 3) = \frac{5 i}{2}$

Step 6.2.7

The final answer is $\frac{5 i}{2}$ .

$\frac{5 i}{2}$

Step 6.3

At $x = - 3$ the derivative is $\frac{5 i}{2}$ . Since this contains an imaginary number, the function does not exist on $(- \infty, - 2)$ .

Function is not real on $(- \infty, - 2)$ since $f' (x)$ is imaginary

Step 7

Substitute a value from the interval $(- 2, - \frac{4}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1.6666667$ in the expression.

$f' (- 1.6666667) = \frac{3 (- 1.6666667) + 4}{2 {((- 1.6666667) + 2)}^{\frac{1}{2}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Multiply $3$ by $- 1.6666667$ .

$f' (- 1.6666667) = \frac{- 5.0000001 + 4}{2 {(- 1.6666667 + 2)}^{\frac{1}{2}}}$

Step 7.2.1.2

Add $- 5.0000001$ and $4$ .

$f' (- 1.6666667) = \frac{- 1.0000001}{2 {(- 1.6666667 + 2)}^{\frac{1}{2}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $- 1.6666667$ and $2$ .

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot {0.3333333}^{\frac{1}{2}}}$

Step 7.2.2.2

Rewrite $0.3333333$ as ${0.57735024}^{2}$ .

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot {({0.57735024}^{2})}^{\frac{1}{2}}}$

Step 7.2.2.3

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

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot {0.57735024}^{2 (\frac{1}{2})}}$

Step 7.2.2.4

Cancel the common factor of $2$ .

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Step 7.2.2.4.1

Cancel the common factor.

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot {0.57735024}^{2 (\frac{1}{2})}}$

Step 7.2.2.4.2

Rewrite the expression.

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot 0.57735024}$

Step 7.2.2.5

Evaluate the exponent.

$f' (- 1.6666667) = \frac{- 1.0000001}{2 \cdot 0.57735024}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $2$ by $0.57735024$ .

$f' (- 1.6666667) = \frac{- 1.0000001}{1.15470048}$

Step 7.2.3.2

Divide $- 1.0000001$ by $1.15470048$ .

$f' (- 1.6666667) = - 0.86602553$

Step 7.2.4

The final answer is $- 0.86602553$ .

$- 0.86602553$

Step 7.3

At $x = - 1.6666667$ the derivative is $- 0.86602553$ . Since this is negative, the function is decreasing on $(- 2, - \frac{4}{3})$ .

Decreasing on $(- 2, - \frac{4}{3})$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- \frac{4}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 0.3333334$ in the expression.

$f' (- 0.3333334) = \frac{3 (- 0.3333334) + 4}{2 {((- 0.3333334) + 2)}^{\frac{1}{2}}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $3$ by $- 0.3333334$ .

$f' (- 0.3333334) = \frac{- 1.0000002 + 4}{2 {(- 0.3333334 + 2)}^{\frac{1}{2}}}$

Step 8.2.1.2

Add $- 1.0000002$ and $4$ .

$f' (- 0.3333334) = \frac{2.9999998}{2 {(- 0.3333334 + 2)}^{\frac{1}{2}}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $- 0.3333334$ and $2$ .

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot {1.6666666}^{\frac{1}{2}}}$

Step 8.2.2.2

Rewrite $1.6666666$ as ${1.29099442}^{2}$ .

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot {({1.29099442}^{2})}^{\frac{1}{2}}}$

Step 8.2.2.3

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

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot {1.29099442}^{2 (\frac{1}{2})}}$

Step 8.2.2.4

Cancel the common factor of $2$ .

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Step 8.2.2.4.1

Cancel the common factor.

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot {1.29099442}^{2 (\frac{1}{2})}}$

Step 8.2.2.4.2

Rewrite the expression.

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot 1.29099442}$

Step 8.2.2.5

Evaluate the exponent.

$f' (- 0.3333334) = \frac{2.9999998}{2 \cdot 1.29099442}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $2$ by $1.29099442$ .

$f' (- 0.3333334) = \frac{2.9999998}{2.58198884}$

Step 8.2.3.2

Divide $2.9999998$ by $2.58198884$ .

$f' (- 0.3333334) = 1.16189494$

Step 8.2.4

The final answer is $1.16189494$ .

$1.16189494$

Step 8.3

At $x = - 0.3333334$ the derivative is $1.16189494$ . Since this is positive, the function is increasing on $(- \frac{4}{3}, \infty)$ .

Increasing on $(- \frac{4}{3}, \infty)$ since $f' (x) > 0$

Step 9

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

Increasing on: $(- \frac{4}{3}, \infty)$

Decreasing on: $(- 2, - \frac{4}{3})$

Step 10