Calculus Examples

$x \sqrt{x + 18}$

Step 1

Write $x \sqrt{x + 18}$ as a function.

$f (x) = x \sqrt{x + 18}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 18}$ as ${(x + 18)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(x + 18)}^{\frac{1}{2}}]$

Step 2.1.2

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 $g (x) = {(x + 18)}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}] + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.3

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 $g (x) = x + 18$ .

Tap for more steps...

Step 2.1.3.1

To apply the Chain Rule, set $u$ as $x + 18$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.3.2

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.3.3

Replace all occurrences of $u$ with $x + 18$ .

$x (\frac{1}{2} {(x + 18)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.4

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

$x (\frac{1}{2} {(x + 18)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.5

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

$x (\frac{1}{2} {(x + 18)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.6

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(x + 18)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.7

Simplify the numerator.

Tap for more steps...

Step 2.1.7.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(x + 18)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.7.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(x + 18)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.8

Combine fractions.

Tap for more steps...

Step 2.1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(x + 18)}^{- \frac{1}{2}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.8.2

Combine $\frac{1}{2}$ and ${(x + 18)}^{- \frac{1}{2}}$ .

$x (\frac{{(x + 18)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.8.3

Move ${(x + 18)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} \frac{d}{d x} [x + 18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.8.4

Combine $\frac{1}{2 {(x + 18)}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} \frac{d}{d x} [x + 18] + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.9

By the Sum Rule, the derivative of $x + 18$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [18]$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.10

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

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [18]) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.11

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} (1 + 0) + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.12

Simplify the expression.

Tap for more steps...

Step 2.1.12.1

Add $1$ and $0$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} \cdot 1 + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.12.2

Multiply $\frac{x}{2 {(x + 18)}^{\frac{1}{2}}}$ by $1$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} + {(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.13

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

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} + {(x + 18)}^{\frac{1}{2}} \cdot 1$

Step 2.1.14

Multiply ${(x + 18)}^{\frac{1}{2}}$ by $1$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} + {(x + 18)}^{\frac{1}{2}}$

Step 2.1.15

To write ${(x + 18)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 18)}^{\frac{1}{2}}}{2 {(x + 18)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} + {(x + 18)}^{\frac{1}{2}} \cdot \frac{2 {(x + 18)}^{\frac{1}{2}}}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.16

Combine ${(x + 18)}^{\frac{1}{2}}$ and $\frac{2 {(x + 18)}^{\frac{1}{2}}}{2 {(x + 18)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 18)}^{\frac{1}{2}}} + \frac{{(x + 18)}^{\frac{1}{2}} (2 {(x + 18)}^{\frac{1}{2}})}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.17

Combine the numerators over the common denominator.

$\frac{x + {(x + 18)}^{\frac{1}{2}} (2 {(x + 18)}^{\frac{1}{2}})}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.18

Multiply ${(x + 18)}^{\frac{1}{2}}$ by ${(x + 18)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 2.1.18.1

Move ${(x + 18)}^{\frac{1}{2}}$ .

$\frac{x + {(x + 18)}^{\frac{1}{2}} {(x + 18)}^{\frac{1}{2}} \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.18.2

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

$\frac{x + {(x + 18)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.18.3

Combine the numerators over the common denominator.

$\frac{x + {(x + 18)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.18.4

Add $1$ and $1$ .

$\frac{x + {(x + 18)}^{\frac{2}{2}} \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.18.5

Divide $2$ by $2$ .

$\frac{x + {(x + 18)}^{1} \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.19

Simplify ${(x + 18)}^{1} \cdot 2$ .

$\frac{x + (x + 18) \cdot 2}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.20

Move $2$ to the left of $x + 18$ .

$\frac{x + 2 \cdot (x + 18)}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21

Simplify.

Tap for more steps...

Step 2.1.21.1

Apply the distributive property.

$\frac{x + 2 x + 2 \cdot 18}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21.2

Simplify the numerator.

Tap for more steps...

Step 2.1.21.2.1

Multiply $2$ by $18$ .

$\frac{x + 2 x + 36}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21.2.2

Add $x$ and $2 x$ .

$\frac{3 x + 36}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21.3

Factor $3$ out of $3 x + 36$ .

Tap for more steps...

Step 2.1.21.3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 36}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21.3.2

Factor $3$ out of $36$ .

$\frac{3 x + 3 \cdot 12}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.1.21.3.3

Factor $3$ out of $3 x + 3 \cdot 12$ .

$f' (x) = \frac{3 (x + 12)}{2 {(x + 18)}^{\frac{1}{2}}}$

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 (x + 12)}{2 {(x + 18)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\frac{x + 12}{{(x + 18)}^{\frac{1}{2}}}]$ .

$\frac{3}{2} \frac{d}{d x} [\frac{x + 12}{{(x + 18)}^{\frac{1}{2}}}]$

Step 2.2.2

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 $f (x) = x + 12$ and $g (x) = {(x + 18)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{{({(x + 18)}^{\frac{1}{2}})}^{2}}$

Step 2.2.3

Multiply the exponents in ${({(x + 18)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.3.1

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{{(x + 18)}^{\frac{1}{2} \cdot 2}}$

Step 2.2.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.2.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{{(x + 18)}^{\frac{1}{2} \cdot 2}}$

Step 2.2.3.2.2

Rewrite the expression.

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{{(x + 18)}^{1}}$

Step 2.2.4

Simplify.

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.5

Differentiate.

Tap for more steps...

Step 2.2.5.1

By the Sum Rule, the derivative of $x + 12$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [12]$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [12]) - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.5.2

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} (1 + \frac{d}{d x} [12]) - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.5.3

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} (1 + 0) - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.5.4

Simplify the expression.

Tap for more steps...

Step 2.2.5.4.1

Add $1$ and $0$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} \cdot 1 - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.5.4.2

Multiply ${(x + 18)}^{\frac{1}{2}}$ by $1$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) \frac{d}{d x} [{(x + 18)}^{\frac{1}{2}}]}{x + 18}$

Step 2.2.6

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 $g (x) = x + 18$ .

Tap for more steps...

Step 2.2.6.1

To apply the Chain Rule, set $u_{1}$ as $x + 18$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.6.2

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.6.3

Replace all occurrences of $u_{1}$ with $x + 18$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.7

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.8

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.9

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.10

Simplify the numerator.

Tap for more steps...

Step 2.2.10.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.10.2

Subtract $2$ from $1$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.11

Combine fractions.

Tap for more steps...

Step 2.2.11.1

Move the negative in front of the fraction.

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2} {(x + 18)}^{- \frac{1}{2}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.11.2

Combine $\frac{1}{2}$ and ${(x + 18)}^{- \frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{{(x + 18)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.11.3

Move ${(x + 18)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} \frac{d}{d x} [x + 18])}{x + 18}$

Step 2.2.12

By the Sum Rule, the derivative of $x + 18$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [18]$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [18]))}{x + 18}$

Step 2.2.13

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

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [18]))}{x + 18}$

Step 2.2.14

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} (1 + 0))}{x + 18}$

Step 2.2.15

Combine fractions.

Tap for more steps...

Step 2.2.15.1

Add $1$ and $0$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) (\frac{1}{2 {(x + 18)}^{\frac{1}{2}}} \cdot 1)}{x + 18}$

Step 2.2.15.2

Multiply $\frac{1}{2 {(x + 18)}^{\frac{1}{2}}}$ by $1$ .

$\frac{3}{2} \cdot \frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}}}{x + 18}$

Step 2.2.15.3

Multiply $\frac{3}{2}$ by $\frac{{(x + 18)}^{\frac{1}{2}} - (x + 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}}}{x + 18}$ .

$\frac{3 ({(x + 18)}^{\frac{1}{2}} - (x + 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 (x + 18)}$

Step 2.2.16

Simplify.

Tap for more steps...

Step 2.2.16.1

Apply the distributive property.

$\frac{3 ({(x + 18)}^{\frac{1}{2}} + (- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 (x + 18)}$

Step 2.2.16.2

Apply the distributive property.

$\frac{3 {(x + 18)}^{\frac{1}{2}} + 3 ((- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 (x + 18)}$

Step 2.2.16.3

Apply the distributive property.

$\frac{3 {(x + 18)}^{\frac{1}{2}} + 3 ((- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 x + 2 \cdot 18}$

Step 2.2.16.4

Simplify the numerator.

Tap for more steps...

Step 2.2.16.4.1

Factor $3$ out of $3 {(x + 18)}^{\frac{1}{2}} + 3 (- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 2.2.16.4.1.1

Factor $3$ out of $3 {(x + 18)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 18)}^{\frac{1}{2}}) + 3 (- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.1.2

Factor $3$ out of $3 (- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}}$ .

$\frac{3 ({(x + 18)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 x + 2 \cdot 18}$

Step 2.2.16.4.1.3

Factor $3$ out of $3 ({(x + 18)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})$ .

$\frac{3 ({(x + 18)}^{\frac{1}{2}} + (- x - 1 \cdot 12) \frac{1}{2 {(x + 18)}^{\frac{1}{2}}})}{2 x + 2 \cdot 18}$

Step 2.2.16.4.2

Let $u_{2} = {(x + 18)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x + 18)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.2.16.4.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{2 u_{2} \cdot u_{2} - x - 12}{2 u_{2}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.2.2

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

Tap for more steps...

Step 2.2.16.4.2.2.1

Move $u_{2}$ .

$\frac{3 \frac{2 (u_{2} \cdot u_{2}) - x - 12}{2 u_{2}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.2.2.2

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

$\frac{3 \frac{2 {u_{2}}^{2} - x - 12}{2 u_{2}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.3

Replace all occurrences of $u_{2}$ with ${(x + 18)}^{\frac{1}{2}}$ .

$\frac{3 \frac{2 {({(x + 18)}^{\frac{1}{2}})}^{2} - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4

Simplify.

Tap for more steps...

Step 2.2.16.4.4.1

Simplify each term.

Tap for more steps...

Step 2.2.16.4.4.1.1

Multiply the exponents in ${({(x + 18)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.16.4.4.1.1.1

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

$\frac{3 \frac{2 {(x + 18)}^{\frac{1}{2} \cdot 2} - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.16.4.4.1.1.2.1

Cancel the common factor.

$\frac{3 \frac{2 {(x + 18)}^{\frac{1}{2} \cdot 2} - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.1.1.2.2

Rewrite the expression.

$\frac{3 \frac{2 {(x + 18)}^{1} - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.1.2

Simplify.

$\frac{3 \frac{2 (x + 18) - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.1.3

Apply the distributive property.

$\frac{3 \frac{2 x + 2 \cdot 18 - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.1.4

Multiply $2$ by $18$ .

$\frac{3 \frac{2 x + 36 - x - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.2

Subtract $x$ from $2 x$ .

$\frac{3 \frac{x + 36 - 12}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.4.4.3

Subtract $12$ from $36$ .

$\frac{3 \frac{x + 24}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.5

Combine terms.

Tap for more steps...

Step 2.2.16.5.1

Combine $3$ and $\frac{x + 24}{2 {(x + 18)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 2 \cdot 18}$

Step 2.2.16.5.2

Multiply $2$ by $18$ .

$\frac{\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 36}$

Step 2.2.16.5.3

Rewrite $\frac{\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}}}}{2 x + 36}$ as a product.

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}}} \cdot \frac{1}{2 x + 36}$

Step 2.2.16.5.4

Multiply $\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}}}$ by $\frac{1}{2 x + 36}$ .

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}} (2 x + 36)}$

Step 2.2.16.6

Simplify the denominator.

Tap for more steps...

Step 2.2.16.6.1

Factor $2$ out of $2 x + 36$ .

Tap for more steps...

Step 2.2.16.6.1.1

Factor $2$ out of $2 x$ .

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}} (2 (x) + 36)}$

Step 2.2.16.6.1.2

Factor $2$ out of $36$ .

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}} (2 x + 2 \cdot 18)}$

Step 2.2.16.6.1.3

Factor $2$ out of $2 x + 2 \cdot 18$ .

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}} (2 (x + 18))}$

$\frac{3 (x + 24)}{2 {(x + 18)}^{\frac{1}{2}} \cdot 2 (x + 18)}$

Step 2.2.16.6.2

Combine exponents.

Tap for more steps...

Step 2.2.16.6.2.1

Multiply $2$ by $2$ .

$\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{1}{2}} (x + 18)}$

Step 2.2.16.6.2.2

Raise $x + 18$ to the power of $1$ .

$\frac{3 (x + 24)}{4 ({(x + 18)}^{1} {(x + 18)}^{\frac{1}{2}})}$

Step 2.2.16.6.2.3

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

$\frac{3 (x + 24)}{4 {(x + 18)}^{1 + \frac{1}{2}}}$

Step 2.2.16.6.2.4

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

$\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.2.16.6.2.5

Combine the numerators over the common denominator.

$\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{2 + 1}{2}}}$

Step 2.2.16.6.2.6

Add $2$ and $1$ .

$f'' (x) = \frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}}$ .

$\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}} = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}} = 0$

Step 3.2

Set the numerator equal to zero.

$3 (x + 24) = 0$

Step 3.3

Solve the equation for $x$ .

Tap for more steps...

Step 3.3.1

Divide each term in $3 (x + 24) = 0$ by $3$ and simplify.

Tap for more steps...

Step 3.3.1.1

Divide each term in $3 (x + 24) = 0$ by $3$ .

$\frac{3 (x + 24)}{3} = \frac{0}{3}$

Step 3.3.1.2

Simplify the left side.

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 24)}{3} = \frac{0}{3}$

Step 3.3.1.2.1.2

Divide $x + 24$ by $1$ .

$x + 24 = \frac{0}{3}$

Step 3.3.1.3

Simplify the right side.

Tap for more steps...

Step 3.3.1.3.1

Divide $0$ by $3$ .

$x + 24 = 0$

Step 3.3.2

Subtract $24$ from both sides of the equation.

$x = - 24$

Step 3.4

Exclude the solutions that do not make $\frac{3 (x + 24)}{4 {(x + 18)}^{\frac{3}{2}}} = 0$ true.

$No solution$

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points