Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Rewrite $\frac{d}{d x} [\sqrt{x^{3} + 3 x^{2}}]$ as $\frac{d}{d x} [x \sqrt{x + 3}]$ .

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Step 1.1.1

Rewrite $x^{3} + 3 x^{2}$ as $x^{2} (x + 3)$ .

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Step 1.1.1.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 1.1.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + x^{2} \cdot 3}]$

Step 1.1.1.3

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

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

Step 1.1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt{x + 3}]$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

To apply the Chain Rule, set $u$ as $x + 3$ .

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

Step 1.4.2

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

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

Step 1.4.3

Replace all occurrences of $u$ with $x + 3$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Combine the numerators over the common denominator.

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

Multiply $- 1$ by $2$ .

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

Step 1.8.2

Subtract $2$ from $1$ .

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

Step 1.9

Combine fractions.

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Step 1.9.1

Move the negative in front of the fraction.

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

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

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

Step 1.10

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 1.11

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

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

Step 1.12

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 1.13

Simplify the expression.

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Step 1.13.1

Add $1$ and $0$ .

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

Step 1.13.2

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

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

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

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

Step 1.18

Combine the numerators over the common denominator.

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

Step 1.19

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

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Step 1.19.1

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

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

Step 1.19.2

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

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

Step 1.19.3

Combine the numerators over the common denominator.

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

Step 1.19.4

Add $1$ and $1$ .

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

Step 1.19.5

Divide $2$ by $2$ .

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

Step 1.20

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

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

Step 1.21

Move $2$ to the left of $x + 3$ .

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

Step 1.22

Simplify.

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Step 1.22.1

Apply the distributive property.

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

Step 1.22.2

Simplify the numerator.

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Step 1.22.2.1

Multiply $2$ by $3$ .

$\frac{x + 2 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.22.2.2

Add $x$ and $2 x$ .

$\frac{3 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.22.3

Factor $3$ out of $3 x + 6$ .

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Step 1.22.3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.22.3.2

Factor $3$ out of $6$ .

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

Step 1.22.3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x + 3)}^{\frac{1}{2}} (1 + 0) - (x + 2) \frac{d}{d x} {(x + 3)}^{\frac{1}{2}}}{x + 3}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

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

Step 2.5.4.2

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

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

Step 2.6

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 $g (x) = x + 3$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{1}$ as $x + 3$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x + 3)}^{\frac{1}{2}} - (x + 2) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (x + 3))}{x + 3}$

Step 2.6.2

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

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

Step 2.6.3

Replace all occurrences of $u_{1}$ with $x + 3$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

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

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

Step 2.11.3

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

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

Step 2.12

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 2.13

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

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

Step 2.14

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 2.15

Combine fractions.

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Step 2.15.1

Add $1$ and $0$ .

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

Step 2.15.2

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

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

Step 2.15.3

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

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

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

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

Step 2.16.2

Apply the distributive property.

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

Step 2.16.3

Apply the distributive property.

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

Step 2.16.4

Simplify the numerator.

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Step 2.16.4.1

Factor $3$ out of $3 {(x + 3)}^{\frac{1}{2}} + 3 (- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ .

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Step 2.16.4.1.1

Factor $3$ out of $3 {(x + 3)}^{\frac{1}{2}}$ .

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

Step 2.16.4.1.2

Factor $3$ out of $3 (- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ .

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

Step 2.16.4.1.3

Factor $3$ out of $3 ({(x + 3)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}})$ .

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

Step 2.16.4.2

Let $u_{2} = {(x + 3)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x + 3)}^{\frac{1}{2}}$ .

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Step 2.16.4.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{3 (\frac{2 u_{2} u_{2} - x - 2}{2 u_{2}})}{2 x + 2 \cdot 3}$

Step 2.16.4.2.2

Multiply $u_{2}$ by $u_{2}$ by adding the exponents.

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Step 2.16.4.2.2.1

Move $u_{2}$ .

$f'' (x) = \frac{3 (\frac{2 (u_{2} u_{2}) - x - 2}{2 u_{2}})}{2 x + 2 \cdot 3}$

Step 2.16.4.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

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

Step 2.16.4.3

Replace all occurrences of $u_{2}$ with ${(x + 3)}^{\frac{1}{2}}$ .

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

Step 2.16.4.4

Simplify.

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Step 2.16.4.4.1

Simplify each term.

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Step 2.16.4.4.1.1

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

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Step 2.16.4.4.1.1.1

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

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

Step 2.16.4.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.16.4.4.1.1.2.1

Cancel the common factor.

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

Step 2.16.4.4.1.1.2.2

Rewrite the expression.

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

Step 2.16.4.4.1.2

Simplify.

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

Step 2.16.4.4.1.3

Apply the distributive property.

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

Step 2.16.4.4.1.4

Multiply $2$ by $3$ .

$f'' (x) = \frac{3 (\frac{2 x + 6 - x - 2}{2 {(x + 3)}^{\frac{1}{2}}})}{2 x + 2 \cdot 3}$

Step 2.16.4.4.2

Subtract $x$ from $2 x$ .

$f'' (x) = \frac{3 (\frac{x + 6 - 2}{2 {(x + 3)}^{\frac{1}{2}}})}{2 x + 2 \cdot 3}$

Step 2.16.4.4.3

Subtract $2$ from $6$ .

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

Step 2.16.5

Combine terms.

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Step 2.16.5.1

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

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

Step 2.16.5.2

Multiply $2$ by $3$ .

$f'' (x) = \frac{\frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}}}{2 x + 6}$

Step 2.16.5.3

Rewrite $\frac{\frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}}}{2 x + 6}$ as a product.

$f'' (x) = \frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}} \cdot \frac{1}{2 x + 6}$

Step 2.16.5.4

Multiply $\frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}}$ by $\frac{1}{2 x + 6}$ .

$f'' (x) = \frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}} (2 x + 6)}$

Step 2.16.6

Simplify the denominator.

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Step 2.16.6.1

Factor $2$ out of $2 x + 6$ .

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Step 2.16.6.1.1

Factor $2$ out of $2 x$ .

$f'' (x) = \frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}} (2 (x) + 6)}$

Step 2.16.6.1.2

Factor $2$ out of $6$ .

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

Step 2.16.6.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

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

Step 2.16.6.2

Combine exponents.

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Step 2.16.6.2.1

Multiply $2$ by $2$ .

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

Step 2.16.6.2.2

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

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

Step 2.16.6.2.3

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

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

Step 2.16.6.2.4

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

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

Step 2.16.6.2.5

Combine the numerators over the common denominator.

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

Step 2.16.6.2.6

Add $2$ and $1$ .

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Rewrite $\frac{d}{d x} [\sqrt{x^{3} + 3 x^{2}}]$ as $\frac{d}{d x} [x \sqrt{x + 3}]$ .

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Step 4.1.1.1

Rewrite $x^{3} + 3 x^{2}$ as $x^{2} (x + 3)$ .

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Step 4.1.1.1.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 4.1.1.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + x^{2} \cdot 3}]$

Step 4.1.1.1.3

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

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

Step 4.1.1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt{x + 3}]$

Step 4.1.2

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

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

Step 4.1.3

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

Step 4.1.4

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

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Step 4.1.4.1

To apply the Chain Rule, set $u$ as $x + 3$ .

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

Step 4.1.4.2

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

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

Step 4.1.4.3

Replace all occurrences of $u$ with $x + 3$ .

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

Combine the numerators over the common denominator.

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

Step 4.1.8

Simplify the numerator.

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Step 4.1.8.1

Multiply $- 1$ by $2$ .

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

Step 4.1.8.2

Subtract $2$ from $1$ .

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

Step 4.1.9

Combine fractions.

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Step 4.1.9.1

Move the negative in front of the fraction.

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

Step 4.1.9.2

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

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

Step 4.1.9.3

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

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

Step 4.1.9.4

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

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

Step 4.1.10

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 4.1.11

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

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

Step 4.1.12

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 4.1.13

Simplify the expression.

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Step 4.1.13.1

Add $1$ and $0$ .

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

Step 4.1.13.2

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

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

Step 4.1.14

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

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

Step 4.1.15

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

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

Step 4.1.16

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

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

Step 4.1.17

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

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

Step 4.1.18

Combine the numerators over the common denominator.

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

Step 4.1.19

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

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Step 4.1.19.1

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

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

Step 4.1.19.2

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

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

Step 4.1.19.3

Combine the numerators over the common denominator.

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

Step 4.1.19.4

Add $1$ and $1$ .

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

Step 4.1.19.5

Divide $2$ by $2$ .

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

Step 4.1.20

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

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

Step 4.1.21

Move $2$ to the left of $x + 3$ .

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

Step 4.1.22

Simplify.

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Step 4.1.22.1

Apply the distributive property.

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

Step 4.1.22.2

Simplify the numerator.

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Step 4.1.22.2.1

Multiply $2$ by $3$ .

$\frac{x + 2 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 4.1.22.2.2

Add $x$ and $2 x$ .

$\frac{3 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 4.1.22.3

Factor $3$ out of $3 x + 6$ .

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Step 4.1.22.3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 4.1.22.3.2

Factor $3$ out of $6$ .

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

Step 4.1.22.3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$3 (x + 2) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

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Step 5.3.1.1

Divide each term in $3 (x + 2) = 0$ by $3$ .

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

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $3$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

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

Step 5.3.1.2.1.2

Divide $x + 2$ by $1$ .

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

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $3$ .

$x + 2 = 0$

Step 5.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

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

Step 6.1.2

Anything raised to $1$ is the base itself.

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

Step 6.2

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

$2 \sqrt{x + 3} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

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

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

Step 6.3.2.2.1.3

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

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Step 6.3.2.2.1.3.1

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

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

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.4

Simplify.

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

Step 6.3.2.2.1.5

Apply the distributive property.

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

Step 6.3.2.2.1.6

Multiply $4$ by $3$ .

$4 x + 12 = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$4 x + 12 = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Subtract $12$ from both sides of the equation.

$4 x = - 12$

Step 6.3.3.2

Divide each term in $4 x = - 12$ by $4$ and simplify.

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Step 6.3.3.2.1

Divide each term in $4 x = - 12$ by $4$ .

$\frac{4 x}{4} = \frac{- 12}{4}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of $4$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 12}{4}$

Step 6.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 12}{4}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 12$ by $4$ .

$x = - 3$

Step 6.4

Set the radicand in $\sqrt{x + 3}$ less than $0$ to find where the expression is undefined.

$x + 3 < 0$

Step 6.5

Subtract $3$ from both sides of the inequality.

$x < - 3$

Step 6.6

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 $0$ .

$x \leq - 3$

$(- \infty, - 3]$

$x \leq - 3$

$(- \infty, - 3]$

Step 7

Critical points to evaluate.

$x = - 2, - 3$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Factor $2$ out of $3 ((- 2) + 4)$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $4 {((- 2) + 3)}^{\frac{3}{2}}$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 9.3

Add $- 1$ and $2$ .

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

Step 9.4

Simplify the denominator.

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Step 9.4.1

Add $- 2$ and $3$ .

$\frac{3 \cdot 1}{2 \cdot 1^{\frac{3}{2}}}$

Step 9.4.2

One to any power is one.

$\frac{3 \cdot 1}{2 \cdot 1}$

Step 9.5

Simplify.

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Step 9.5.1

Multiply $3$ by $1$ .

$\frac{3}{2 \cdot 1}$

Step 9.5.2

Multiply $2$ by $1$ .

$\frac{3}{2}$

Step 10

$x = - 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 2$ is a local minimum

Step 11

Find the y-value when $x = - 2$ .

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Step 11.1

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

$f (- 2) = \sqrt{{(- 2)}^{3} + 3 {(- 2)}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Raise $- 2$ to the power of $3$ .

$f (- 2) = \sqrt{- 8 + 3 {(- 2)}^{2}}$

Step 11.2.2

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

$f (- 2) = \sqrt{- 8 + 3 \cdot 4}$

Step 11.2.3

Multiply $3$ by $4$ .

$f (- 2) = \sqrt{- 8 + 12}$

Step 11.2.4

Add $- 8$ and $12$ .

$f (- 2) = \sqrt{4}$

Step 11.2.5

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

$f (- 2) = \sqrt{2^{2}}$

Step 11.2.6

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

$f (- 2) = 2$

Step 11.2.7

The final answer is $2$ .

$y = 2$

Step 12

Evaluate the second derivative at $x = - 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Add $- 3$ and $3$ .

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

Step 13.1.2

Rewrite $0$ as $0^{2}$ .

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

Step 13.1.3

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

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

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

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

Step 13.2.2

Rewrite the expression.

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

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$\frac{3 ((- 3) + 4)}{4 \cdot 0}$

Step 13.3.2

Multiply $4$ by $0$ .

$\frac{3 ((- 3) + 4)}{0}$

Step 13.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{3 ((- 3) + 4)}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15