Calculus Examples

$y = x \sqrt{x + 9}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(x + 9)}^{\frac{1}{2}}]$

Step 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 $g (x) = {(x + 9)}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}] + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 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 $g (x) = x + 9$ .

Tap for more steps...

Step 1.3.1

To apply the Chain Rule, set $u$ as $x + 9$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.3.2

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.3.3

Replace all occurrences of $u$ with $x + 9$ .

$x (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.4

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

$x (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.5

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

$x (\frac{1}{2} {(x + 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.6

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(x + 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.7

Simplify the numerator.

Tap for more steps...

Step 1.7.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(x + 9)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.7.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(x + 9)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8

Combine fractions.

Tap for more steps...

Step 1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(x + 9)}^{- \frac{1}{2}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(x + 9)}^{- \frac{1}{2}}$ .

$x (\frac{{(x + 9)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.3

Move ${(x + 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.4

Combine $\frac{1}{2 {(x + 9)}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9] + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.9

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.10

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

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [9]) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.11

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} (1 + 0) + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.12

Simplify the expression.

Tap for more steps...

Step 1.12.1

Add $1$ and $0$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} \cdot 1 + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.12.2

Multiply $\frac{x}{2 {(x + 9)}^{\frac{1}{2}}}$ by $1$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} + {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.13

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

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} + {(x + 9)}^{\frac{1}{2}} \cdot 1$

Step 1.14

Multiply ${(x + 9)}^{\frac{1}{2}}$ by $1$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} + {(x + 9)}^{\frac{1}{2}}$

Step 1.15

To write ${(x + 9)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 9)}^{\frac{1}{2}}}{2 {(x + 9)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} + {(x + 9)}^{\frac{1}{2}} \cdot \frac{2 {(x + 9)}^{\frac{1}{2}}}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.16

Combine ${(x + 9)}^{\frac{1}{2}}$ and $\frac{2 {(x + 9)}^{\frac{1}{2}}}{2 {(x + 9)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 9)}^{\frac{1}{2}}} + \frac{{(x + 9)}^{\frac{1}{2}} (2 {(x + 9)}^{\frac{1}{2}})}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.17

Combine the numerators over the common denominator.

$\frac{x + {(x + 9)}^{\frac{1}{2}} (2 {(x + 9)}^{\frac{1}{2}})}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.18

Multiply ${(x + 9)}^{\frac{1}{2}}$ by ${(x + 9)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 1.18.1

Move ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{x + {(x + 9)}^{\frac{1}{2}} {(x + 9)}^{\frac{1}{2}} \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.18.2

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

$\frac{x + {(x + 9)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.18.3

Combine the numerators over the common denominator.

$\frac{x + {(x + 9)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.18.4

Add $1$ and $1$ .

$\frac{x + {(x + 9)}^{\frac{2}{2}} \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.18.5

Divide $2$ by $2$ .

$\frac{x + {(x + 9)}^{1} \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.19

Simplify ${(x + 9)}^{1} \cdot 2$ .

$\frac{x + (x + 9) \cdot 2}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.20

Move $2$ to the left of $x + 9$ .

$\frac{x + 2 \cdot (x + 9)}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21

Simplify.

Tap for more steps...

Step 1.21.1

Apply the distributive property.

$\frac{x + 2 x + 2 \cdot 9}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21.2

Simplify the numerator.

Tap for more steps...

Step 1.21.2.1

Multiply $2$ by $9$ .

$\frac{x + 2 x + 18}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21.2.2

Add $x$ and $2 x$ .

$\frac{3 x + 18}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21.3

Factor $3$ out of $3 x + 18$ .

Tap for more steps...

Step 1.21.3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 18}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21.3.2

Factor $3$ out of $18$ .

$\frac{3 x + 3 \cdot 6}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 1.21.3.3

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

$f' (x) = \frac{3 (x + 6)}{2 {(x + 9)}^{\frac{1}{2}}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 (x + 6)}{2 {(x + 9)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\frac{x + 6}{{(x + 9)}^{\frac{1}{2}}}]$ .

$\frac{3}{2} \frac{d}{d x} [\frac{x + 6}{{(x + 9)}^{\frac{1}{2}}}]$

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 $f (x) = x + 6$ and $g (x) = {(x + 9)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{{({(x + 9)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.3.1

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{{(x + 9)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{{(x + 9)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{{(x + 9)}^{1}}$

Step 2.4

Simplify.

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [6]) - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

Step 2.5.2

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} (1 + \frac{d}{d x} [6]) - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

Step 2.5.3

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} (1 + 0) - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

Step 2.5.4

Simplify the expression.

Tap for more steps...

Step 2.5.4.1

Add $1$ and $0$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} \cdot 1 - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

Step 2.5.4.2

Multiply ${(x + 9)}^{\frac{1}{2}}$ by $1$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]}{x + 9}$

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 $g (x) = x + 9$ .

Tap for more steps...

Step 2.6.1

To apply the Chain Rule, set $u_{1}$ as $x + 9$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [x + 9])}{x + 9}$

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 $n = \frac{1}{2}$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.6.3

Replace all occurrences of $u_{1}$ with $x + 9$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.7

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.8

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.10.2

Subtract $2$ from $1$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.11

Combine fractions.

Tap for more steps...

Step 2.11.1

Move the negative in front of the fraction.

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2} {(x + 9)}^{- \frac{1}{2}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(x + 9)}^{- \frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{{(x + 9)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.11.3

Move ${(x + 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9])}{x + 9}$

Step 2.12

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [9]))}{x + 9}$

Step 2.13

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

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [9]))}{x + 9}$

Step 2.14

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} (1 + 0))}{x + 9}$

Step 2.15

Combine fractions.

Tap for more steps...

Step 2.15.1

Add $1$ and $0$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} \cdot 1)}{x + 9}$

Step 2.15.2

Multiply $\frac{1}{2 {(x + 9)}^{\frac{1}{2}}}$ by $1$ .

$\frac{3}{2} \cdot \frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}}}{x + 9}$

Step 2.15.3

Multiply $\frac{3}{2}$ by $\frac{{(x + 9)}^{\frac{1}{2}} - (x + 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}}}{x + 9}$ .

$\frac{3 ({(x + 9)}^{\frac{1}{2}} - (x + 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 (x + 9)}$

Step 2.16

Simplify.

Tap for more steps...

Step 2.16.1

Apply the distributive property.

$\frac{3 ({(x + 9)}^{\frac{1}{2}} + (- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 (x + 9)}$

Step 2.16.2

Apply the distributive property.

$\frac{3 {(x + 9)}^{\frac{1}{2}} + 3 ((- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 (x + 9)}$

Step 2.16.3

Apply the distributive property.

$\frac{3 {(x + 9)}^{\frac{1}{2}} + 3 ((- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 x + 2 \cdot 9}$

Step 2.16.4

Simplify the numerator.

Tap for more steps...

Step 2.16.4.1

Factor $3$ out of $3 {(x + 9)}^{\frac{1}{2}} + 3 (- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 2.16.4.1.1

Factor $3$ out of $3 {(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{1}{2}}) + 3 (- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.1.2

Factor $3$ out of $3 (- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}}$ .

$\frac{3 ({(x + 9)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 x + 2 \cdot 9}$

Step 2.16.4.1.3

Factor $3$ out of $3 ({(x + 9)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})$ .

$\frac{3 ({(x + 9)}^{\frac{1}{2}} + (- x - 1 \cdot 6) \frac{1}{2 {(x + 9)}^{\frac{1}{2}}})}{2 x + 2 \cdot 9}$

Step 2.16.4.2

Let $u_{2} = {(x + 9)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x + 9)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.16.4.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{2 u_{2} \cdot u_{2} - x - 6}{2 u_{2}}}{2 x + 2 \cdot 9}$

Step 2.16.4.2.2

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

Tap for more steps...

Step 2.16.4.2.2.1

Move $u_{2}$ .

$\frac{3 \frac{2 (u_{2} \cdot u_{2}) - x - 6}{2 u_{2}}}{2 x + 2 \cdot 9}$

Step 2.16.4.2.2.2

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

$\frac{3 \frac{2 {u_{2}}^{2} - x - 6}{2 u_{2}}}{2 x + 2 \cdot 9}$

Step 2.16.4.3

Replace all occurrences of $u_{2}$ with ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 \frac{2 {({(x + 9)}^{\frac{1}{2}})}^{2} - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4

Simplify.

Tap for more steps...

Step 2.16.4.4.1

Simplify each term.

Tap for more steps...

Step 2.16.4.4.1.1

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.16.4.4.1.1.1

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

$\frac{3 \frac{2 {(x + 9)}^{\frac{1}{2} \cdot 2} - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.16.4.4.1.1.2.1

Cancel the common factor.

$\frac{3 \frac{2 {(x + 9)}^{\frac{1}{2} \cdot 2} - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.1.1.2.2

Rewrite the expression.

$\frac{3 \frac{2 {(x + 9)}^{1} - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.1.2

Simplify.

$\frac{3 \frac{2 (x + 9) - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.1.3

Apply the distributive property.

$\frac{3 \frac{2 x + 2 \cdot 9 - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.1.4

Multiply $2$ by $9$ .

$\frac{3 \frac{2 x + 18 - x - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.2

Subtract $x$ from $2 x$ .

$\frac{3 \frac{x + 18 - 6}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.4.4.3

Subtract $6$ from $18$ .

$\frac{3 \frac{x + 12}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.5

Combine terms.

Tap for more steps...

Step 2.16.5.1

Combine $3$ and $\frac{x + 12}{2 {(x + 9)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 2 \cdot 9}$

Step 2.16.5.2

Multiply $2$ by $9$ .

$\frac{\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 18}$

Step 2.16.5.3

Rewrite $\frac{\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}}}}{2 x + 18}$ as a product.

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}}} \cdot \frac{1}{2 x + 18}$

Step 2.16.5.4

Multiply $\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}}}$ by $\frac{1}{2 x + 18}$ .

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}} (2 x + 18)}$

Step 2.16.6

Simplify the denominator.

Tap for more steps...

Step 2.16.6.1

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

Tap for more steps...

Step 2.16.6.1.1

Factor $2$ out of $2 x$ .

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}} (2 (x) + 18)}$

Step 2.16.6.1.2

Factor $2$ out of $18$ .

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}} (2 x + 2 \cdot 9)}$

Step 2.16.6.1.3

Factor $2$ out of $2 x + 2 \cdot 9$ .

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}} (2 (x + 9))}$

$\frac{3 (x + 12)}{2 {(x + 9)}^{\frac{1}{2}} \cdot 2 (x + 9)}$

Step 2.16.6.2

Combine exponents.

Tap for more steps...

Step 2.16.6.2.1

Multiply $2$ by $2$ .

$\frac{3 (x + 12)}{4 {(x + 9)}^{\frac{1}{2}} (x + 9)}$

Step 2.16.6.2.2

Raise $x + 9$ to the power of $1$ .

$\frac{3 (x + 12)}{4 ({(x + 9)}^{1} {(x + 9)}^{\frac{1}{2}})}$

Step 2.16.6.2.3

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

$\frac{3 (x + 12)}{4 {(x + 9)}^{1 + \frac{1}{2}}}$

Step 2.16.6.2.4

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

$\frac{3 (x + 12)}{4 {(x + 9)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.16.6.2.5

Combine the numerators over the common denominator.

$\frac{3 (x + 12)}{4 {(x + 9)}^{\frac{2 + 1}{2}}}$

Step 2.16.6.2.6

Add $2$ and $1$ .

$f'' (x) = \frac{3 (x + 12)}{4 {(x + 9)}^{\frac{3}{2}}}$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3 (x + 12)}{4 {(x + 9)}^{\frac{3}{2}}}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [\frac{x + 12}{{(x + 9)}^{\frac{3}{2}}}]$ .

$\frac{3}{4} \frac{d}{d x} [\frac{x + 12}{{(x + 9)}^{\frac{3}{2}}}]$

Step 3.2

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{({(x + 9)}^{\frac{3}{2}})}^{2}}$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

Multiply the exponents in ${({(x + 9)}^{\frac{3}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.1.1

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{\frac{3}{2} \cdot 2}}$

Step 3.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{\frac{3}{2} \cdot 2}}$

Step 3.3.1.2.2

Rewrite the expression.

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} \frac{d}{d x} [x + 12] - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.3.2

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [12]) - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.3.3

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} (1 + \frac{d}{d x} [12]) - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.3.4

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} (1 + 0) - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.3.5

Simplify the expression.

Tap for more steps...

Step 3.3.5.1

Add $1$ and $0$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} \cdot 1 - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.3.5.2

Multiply ${(x + 9)}^{\frac{3}{2}}$ by $1$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) \frac{d}{d x} [{(x + 9)}^{\frac{3}{2}}]}{{(x + 9)}^{3}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x + 9$ .

Tap for more steps...

Step 3.4.1

To apply the Chain Rule, set $u$ as $x + 9$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.4.3

Replace all occurrences of $u$ with $x + 9$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{3}{2} - 1} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.5

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.6

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.8

Simplify the numerator.

Tap for more steps...

Step 3.8.1

Multiply $- 1$ by $2$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{3 - 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.8.2

Subtract $2$ from $3$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3}{2} {(x + 9)}^{\frac{1}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.9

Combine $\frac{3}{2}$ and ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3 {(x + 9)}^{\frac{1}{2}}}{2} \frac{d}{d x} [x + 9])}{{(x + 9)}^{3}}$

Step 3.10

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3 {(x + 9)}^{\frac{1}{2}}}{2} (\frac{d}{d x} [x] + \frac{d}{d x} [9]))}{{(x + 9)}^{3}}$

Step 3.11

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

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3 {(x + 9)}^{\frac{1}{2}}}{2} (1 + \frac{d}{d x} [9]))}{{(x + 9)}^{3}}$

Step 3.12

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3 {(x + 9)}^{\frac{1}{2}}}{2} (1 + 0))}{{(x + 9)}^{3}}$

Step 3.13

Combine fractions.

Tap for more steps...

Step 3.13.1

Add $1$ and $0$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) (\frac{3 {(x + 9)}^{\frac{1}{2}}}{2} \cdot 1)}{{(x + 9)}^{3}}$

Step 3.13.2

Multiply $\frac{3 {(x + 9)}^{\frac{1}{2}}}{2}$ by $1$ .

$\frac{3}{4} \cdot \frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2}}{{(x + 9)}^{3}}$

Step 3.13.3

Multiply $\frac{3}{4}$ by $\frac{{(x + 9)}^{\frac{3}{2}} - (x + 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2}}{{(x + 9)}^{3}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - (x + 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14

Simplify.

Tap for more steps...

Step 3.14.1

Apply the distributive property.

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + (- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.2

Apply the distributive property.

$\frac{3 {(x + 9)}^{\frac{3}{2}} + 3 ((- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3

Simplify the numerator.

Tap for more steps...

Step 3.14.3.1

Factor $3$ out of $3 {(x + 9)}^{\frac{3}{2}} + 3 (- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2}$ .

Tap for more steps...

Step 3.14.3.1.1

Factor $3$ out of $3 {(x + 9)}^{\frac{3}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}}) + 3 (- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.1.2

Factor $3$ out of $3 (- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}}) + 3 ((- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.1.3

Factor $3$ out of $3 ({(x + 9)}^{\frac{3}{2}}) + 3 ((- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + (- x - 1 \cdot 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.2

Multiply $- 1$ by $12$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + (- x - 12) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.3

Apply the distributive property.

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - x \frac{3 {(x + 9)}^{\frac{1}{2}}}{2} - 12 \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.4

Combine $\frac{3 {(x + 9)}^{\frac{1}{2}}}{2}$ and $x$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} - 12 \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.14.3.5.1

Factor $2$ out of $- 12$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} + 2 (- 6) \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.5.2

Cancel the common factor.

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} + 2 \cdot - 6 \frac{3 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.5.3

Rewrite the expression.

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} - 6 (3 {(x + 9)}^{\frac{1}{2}}))}{4 {(x + 9)}^{3}}$

Step 3.14.3.6

Multiply $3$ by $- 6$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} - 18 {(x + 9)}^{\frac{1}{2}})}{4 {(x + 9)}^{3}}$

Step 3.14.3.7

To write $- 18 {(x + 9)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} - 18 {(x + 9)}^{\frac{1}{2}} \cdot \frac{2}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.8

Combine $- 18 {(x + 9)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} - \frac{3 {(x + 9)}^{\frac{1}{2}} x}{2} + \frac{- 18 {(x + 9)}^{\frac{1}{2}} \cdot 2}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.9

Combine the numerators over the common denominator.

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{- 3 {(x + 9)}^{\frac{1}{2}} x - 18 {(x + 9)}^{\frac{1}{2}} \cdot 2}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10

Simplify the numerator.

Tap for more steps...

Step 3.14.3.10.1

Factor $3 {(x + 9)}^{\frac{1}{2}}$ out of $- 3 {(x + 9)}^{\frac{1}{2}} x - 18 {(x + 9)}^{\frac{1}{2}} \cdot 2$ .

Tap for more steps...

Step 3.14.3.10.1.1

Reorder the expression.

Tap for more steps...

Step 3.14.3.10.1.1.1

Move ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{- 3 x {(x + 9)}^{\frac{1}{2}} - 18 {(x + 9)}^{\frac{1}{2}} \cdot 2}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10.1.1.2

Move ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{- 3 x {(x + 9)}^{\frac{1}{2}} - 18 \cdot 2 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10.1.2

Factor $3 {(x + 9)}^{\frac{1}{2}}$ out of $- 3 x {(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x) - 18 \cdot 2 {(x + 9)}^{\frac{1}{2}}}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10.1.3

Factor $3 {(x + 9)}^{\frac{1}{2}}$ out of $- 18 \cdot 2 {(x + 9)}^{\frac{1}{2}}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x) + 3 {(x + 9)}^{\frac{1}{2}} (- 6 \cdot 2)}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10.1.4

Factor $3 {(x + 9)}^{\frac{1}{2}}$ out of $3 {(x + 9)}^{\frac{1}{2}} (- x) + 3 {(x + 9)}^{\frac{1}{2}} (- 6 \cdot 2)$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 6 \cdot 2)}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.10.2

Multiply $- 6$ by $2$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.11

To write ${(x + 9)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 ({(x + 9)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.12

Combine ${(x + 9)}^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (\frac{{(x + 9)}^{\frac{3}{2}} \cdot 2}{2} + \frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2})}{4 {(x + 9)}^{3}}$

Step 3.14.3.13

Combine the numerators over the common denominator.

$\frac{3 \frac{{(x + 9)}^{\frac{3}{2}} \cdot 2 + 3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14

Rewrite $\frac{{(x + 9)}^{\frac{3}{2}} \cdot 2 + 3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2}$ in a factored form.

Tap for more steps...

Step 3.14.3.14.1

Factor ${(x + 9)}^{\frac{1}{2}}$ out of ${(x + 9)}^{\frac{3}{2}} \cdot 2 + 3 {(x + 9)}^{\frac{1}{2}} (- x - 12)$ .

Tap for more steps...

Step 3.14.3.14.1.1

Reorder ${(x + 9)}^{\frac{3}{2}}$ and $2$ .

$\frac{3 \frac{2 \cdot {(x + 9)}^{\frac{3}{2}} + 3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.1.2

Factor ${(x + 9)}^{\frac{1}{2}}$ out of $2 \cdot {(x + 9)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + 3 {(x + 9)}^{\frac{1}{2}} (- x - 12)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.1.3

Factor ${(x + 9)}^{\frac{1}{2}}$ out of $3 {(x + 9)}^{\frac{1}{2}} (- x - 12)$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + {(x + 9)}^{\frac{1}{2}} (3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.1.4

Factor ${(x + 9)}^{\frac{1}{2}}$ out of ${(x + 9)}^{\frac{1}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + {(x + 9)}^{\frac{1}{2}} (3 (- x - 12))$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}} + 3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.2

Divide $2$ by $2$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 \cdot {(x + 9)}^{1} + 3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.3

Simplify.

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 \cdot (x + 9) + 3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.4

Apply the distributive property.

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 x + 2 \cdot 9 + 3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.5

Multiply $2$ by $9$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 x + 18 + 3 (- x - 12))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.6

Apply the distributive property.

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 x + 18 + 3 (- x) + 3 \cdot - 12)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.7

Multiply $- 1$ by $3$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 x + 18 - 3 x + 3 \cdot - 12)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.8

Multiply $3$ by $- 12$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (2 x + 18 - 3 x - 36)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.9

Subtract $3 x$ from $2 x$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (- x + 18 - 36)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.3.14.10

Subtract $36$ from $18$ .

$\frac{3 \frac{{(x + 9)}^{\frac{1}{2}} (- x - 18)}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.4

Combine terms.

Tap for more steps...

Step 3.14.4.1

Combine $3$ and $\frac{{(x + 9)}^{\frac{1}{2}} (- x - 18)}{2}$ .

$\frac{\frac{3 ({(x + 9)}^{\frac{1}{2}} (- x - 18))}{2}}{4 {(x + 9)}^{3}}$

Step 3.14.4.2

Rewrite $\frac{\frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 18)}{2}}{4 {(x + 9)}^{3}}$ as a product.

$\frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 18)}{2} \cdot \frac{1}{4 {(x + 9)}^{3}}$

Step 3.14.4.3

Multiply $\frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 18)}{2}$ by $\frac{1}{4 {(x + 9)}^{3}}$ .

$\frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 18)}{2 (4 {(x + 9)}^{3})}$

Step 3.14.4.4

Multiply $4$ by $2$ .

$\frac{3 {(x + 9)}^{\frac{1}{2}} (- x - 18)}{8 {(x + 9)}^{3}}$

Step 3.14.4.5

Move ${(x + 9)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3 (- x - 18)}{8 {(x + 9)}^{3} {(x + 9)}^{- \frac{1}{2}}}$

Step 3.14.4.6

Multiply ${(x + 9)}^{3}$ by ${(x + 9)}^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 3.14.4.6.1

Move ${(x + 9)}^{- \frac{1}{2}}$ .

$\frac{3 (- x - 18)}{8 ({(x + 9)}^{- \frac{1}{2}} {(x + 9)}^{3})}$

Step 3.14.4.6.2

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

$\frac{3 (- x - 18)}{8 {(x + 9)}^{- \frac{1}{2} + 3}}$

Step 3.14.4.6.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (- x - 18)}{8 {(x + 9)}^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Step 3.14.4.6.4

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 (- x - 18)}{8 {(x + 9)}^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Step 3.14.4.6.5

Combine the numerators over the common denominator.

$\frac{3 (- x - 18)}{8 {(x + 9)}^{\frac{- 1 + 3 \cdot 2}{2}}}$

Step 3.14.4.6.6

Simplify the numerator.

Tap for more steps...

Step 3.14.4.6.6.1

Multiply $3$ by $2$ .

$\frac{3 (- x - 18)}{8 {(x + 9)}^{\frac{- 1 + 6}{2}}}$

Step 3.14.4.6.6.2

Add $- 1$ and $6$ .

$\frac{3 (- x - 18)}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 3.14.5

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) - 18)}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 3.14.6

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{3 (- (x) - 1 (18))}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 3.14.7

Factor $- 1$ out of $- (x) - 1 (18)$ .

$\frac{3 (- (x + 18))}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 3.14.8

Rewrite $- (x + 18)$ as $- 1 (x + 18)$ .

$\frac{3 (- 1 (x + 18))}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 3.14.9

Move the negative in front of the fraction.

$f''' (x) = - \frac{3 (x + 18)}{8 {(x + 9)}^{\frac{5}{2}}}$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

Since $- \frac{3}{8}$ is constant with respect to $x$ , the derivative of $- \frac{3 (x + 18)}{8 {(x + 9)}^{\frac{5}{2}}}$ with respect to $x$ is $- \frac{3}{8} \frac{d}{d x} [\frac{x + 18}{{(x + 9)}^{\frac{5}{2}}}]$ .

$- \frac{3}{8} \frac{d}{d x} [\frac{x + 18}{{(x + 9)}^{\frac{5}{2}}}]$

Step 4.2

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} \frac{d}{d x} [x + 18] - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{({(x + 9)}^{\frac{5}{2}})}^{2}}$

Step 4.3

Differentiate.

Tap for more steps...

Step 4.3.1

Multiply the exponents in ${({(x + 9)}^{\frac{5}{2}})}^{2}$ .

Tap for more steps...

Step 4.3.1.1

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} \frac{d}{d x} [x + 18] - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{\frac{5}{2} \cdot 2}}$

Step 4.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.1.2.1

Cancel the common factor.

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} \frac{d}{d x} [x + 18] - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{\frac{5}{2} \cdot 2}}$

Step 4.3.1.2.2

Rewrite the expression.

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} \frac{d}{d x} [x + 18] - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.3.2

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [18]) - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.3.3

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} (1 + \frac{d}{d x} [18]) - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.3.4

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} (1 + 0) - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.3.5

Simplify the expression.

Tap for more steps...

Step 4.3.5.1

Add $1$ and $0$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} \cdot 1 - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.3.5.2

Multiply ${(x + 9)}^{\frac{5}{2}}$ by $1$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{d}{d x} [{(x + 9)}^{\frac{5}{2}}]}{{(x + 9)}^{5}}$

Step 4.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{5}{2}}$ and $g (x) = x + 9$ .

Tap for more steps...

Step 4.4.1

To apply the Chain Rule, set $u$ as $x + 9$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{d}{d u} [u^{\frac{5}{2}}] \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{5}{2}$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} u^{\frac{5}{2} - 1} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.4.3

Replace all occurrences of $u$ with $x + 9$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{5}{2} - 1} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.5

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{5}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.6

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.7

Combine the numerators over the common denominator.

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{5 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.8

Simplify the numerator.

Tap for more steps...

Step 4.8.1

Multiply $- 1$ by $2$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{5 - 2}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.8.2

Subtract $2$ from $5$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5}{2} {(x + 9)}^{\frac{3}{2}} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.9

Combine $\frac{5}{2}$ and ${(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5 {(x + 9)}^{\frac{3}{2}}}{2} \frac{d}{d x} [x + 9])}{{(x + 9)}^{5}}$

Step 4.10

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5 {(x + 9)}^{\frac{3}{2}}}{2} (\frac{d}{d x} [x] + \frac{d}{d x} [9]))}{{(x + 9)}^{5}}$

Step 4.11

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

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5 {(x + 9)}^{\frac{3}{2}}}{2} (1 + \frac{d}{d x} [9]))}{{(x + 9)}^{5}}$

Step 4.12

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5 {(x + 9)}^{\frac{3}{2}}}{2} (1 + 0))}{{(x + 9)}^{5}}$

Step 4.13

Combine fractions.

Tap for more steps...

Step 4.13.1

Add $1$ and $0$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) (\frac{5 {(x + 9)}^{\frac{3}{2}}}{2} \cdot 1)}{{(x + 9)}^{5}}$

Step 4.13.2

Multiply $\frac{5 {(x + 9)}^{\frac{3}{2}}}{2}$ by $1$ .

$- \frac{3}{8} \cdot \frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}}{{(x + 9)}^{5}}$

Step 4.13.3

Multiply $\frac{{(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}}{{(x + 9)}^{5}}$ by $\frac{3}{8}$ .

$- \frac{({(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}) \cdot 3}{{(x + 9)}^{5} \cdot 8}$

Step 4.13.4

Reorder.

Tap for more steps...

Step 4.13.4.1

Move $3$ to the left of ${(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}$ .

$- \frac{3 \cdot ({(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{{(x + 9)}^{5} \cdot 8}$

Step 4.13.4.2

Move $8$ to the left of ${(x + 9)}^{5}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - (x + 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 \cdot {(x + 9)}^{5}}$

Step 4.14

Simplify.

Tap for more steps...

Step 4.14.1

Apply the distributive property.

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + (- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.2

Apply the distributive property.

$- \frac{3 {(x + 9)}^{\frac{5}{2}} + 3 ((- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3

Simplify the numerator.

Tap for more steps...

Step 4.14.3.1

Factor $3$ out of $3 {(x + 9)}^{\frac{5}{2}} + 3 (- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}$ .

Tap for more steps...

Step 4.14.3.1.1

Factor $3$ out of $3 {(x + 9)}^{\frac{5}{2}}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}}) + 3 (- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.1.2

Factor $3$ out of $3 (- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}}) + 3 ((- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.1.3

Factor $3$ out of $3 ({(x + 9)}^{\frac{5}{2}}) + 3 ((- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + (- x - 1 \cdot 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.2

Multiply $- 1$ by $18$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + (- x - 18) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.3

Apply the distributive property.

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - x \frac{5 {(x + 9)}^{\frac{3}{2}}}{2} - 18 \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.4

Combine $\frac{5 {(x + 9)}^{\frac{3}{2}}}{2}$ and $x$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} - 18 \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.14.3.5.1

Factor $2$ out of $- 18$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} + 2 (- 9) \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.5.2

Cancel the common factor.

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} + 2 \cdot - 9 \frac{5 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.5.3

Rewrite the expression.

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} - 9 (5 {(x + 9)}^{\frac{3}{2}}))}{8 {(x + 9)}^{5}}$

Step 4.14.3.6

Multiply $5$ by $- 9$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} - 45 {(x + 9)}^{\frac{3}{2}})}{8 {(x + 9)}^{5}}$

Step 4.14.3.7

To write $- 45 {(x + 9)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} - 45 {(x + 9)}^{\frac{3}{2}} \cdot \frac{2}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.8

Combine $- 45 {(x + 9)}^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} - \frac{5 {(x + 9)}^{\frac{3}{2}} x}{2} + \frac{- 45 {(x + 9)}^{\frac{3}{2}} \cdot 2}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.9

Combine the numerators over the common denominator.

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{- 5 {(x + 9)}^{\frac{3}{2}} x - 45 {(x + 9)}^{\frac{3}{2}} \cdot 2}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10

Simplify the numerator.

Tap for more steps...

Step 4.14.3.10.1

Factor $5 {(x + 9)}^{\frac{3}{2}}$ out of $- 5 {(x + 9)}^{\frac{3}{2}} x - 45 {(x + 9)}^{\frac{3}{2}} \cdot 2$ .

Tap for more steps...

Step 4.14.3.10.1.1

Reorder the expression.

Tap for more steps...

Step 4.14.3.10.1.1.1

Move ${(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{- 5 x {(x + 9)}^{\frac{3}{2}} - 45 {(x + 9)}^{\frac{3}{2}} \cdot 2}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10.1.1.2

Move ${(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{- 5 x {(x + 9)}^{\frac{3}{2}} - 45 \cdot 2 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10.1.2

Factor $5 {(x + 9)}^{\frac{3}{2}}$ out of $- 5 x {(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x) - 45 \cdot 2 {(x + 9)}^{\frac{3}{2}}}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10.1.3

Factor $5 {(x + 9)}^{\frac{3}{2}}$ out of $- 45 \cdot 2 {(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x) + 5 {(x + 9)}^{\frac{3}{2}} (- 9 \cdot 2)}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10.1.4

Factor $5 {(x + 9)}^{\frac{3}{2}}$ out of $5 {(x + 9)}^{\frac{3}{2}} (- x) + 5 {(x + 9)}^{\frac{3}{2}} (- 9 \cdot 2)$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x - 9 \cdot 2)}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.10.2

Multiply $- 9$ by $2$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.11

To write ${(x + 9)}^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 ({(x + 9)}^{\frac{5}{2}} \cdot \frac{2}{2} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.12

Combine ${(x + 9)}^{\frac{5}{2}}$ and $\frac{2}{2}$ .

$- \frac{3 (\frac{{(x + 9)}^{\frac{5}{2}} \cdot 2}{2} + \frac{5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2})}{8 {(x + 9)}^{5}}$

Step 4.14.3.13

Combine the numerators over the common denominator.

$- \frac{3 \frac{{(x + 9)}^{\frac{5}{2}} \cdot 2 + 5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14

Rewrite $\frac{{(x + 9)}^{\frac{5}{2}} \cdot 2 + 5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2}$ in a factored form.

Tap for more steps...

Step 4.14.3.14.1

Factor ${(x + 9)}^{\frac{3}{2}}$ out of ${(x + 9)}^{\frac{5}{2}} \cdot 2 + 5 {(x + 9)}^{\frac{3}{2}} (- x - 18)$ .

Tap for more steps...

Step 4.14.3.14.1.1

Reorder ${(x + 9)}^{\frac{5}{2}}$ and $2$ .

$- \frac{3 \frac{2 \cdot {(x + 9)}^{\frac{5}{2}} + 5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.1.2

Factor ${(x + 9)}^{\frac{3}{2}}$ out of $2 \cdot {(x + 9)}^{\frac{5}{2}}$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + 5 {(x + 9)}^{\frac{3}{2}} (- x - 18)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.1.3

Factor ${(x + 9)}^{\frac{3}{2}}$ out of $5 {(x + 9)}^{\frac{3}{2}} (- x - 18)$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + {(x + 9)}^{\frac{3}{2}} (5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.1.4

Factor ${(x + 9)}^{\frac{3}{2}}$ out of ${(x + 9)}^{\frac{3}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}}) + {(x + 9)}^{\frac{3}{2}} (5 (- x - 18))$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 \cdot {(x + 9)}^{\frac{2}{2}} + 5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.2

Divide $2$ by $2$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 \cdot {(x + 9)}^{1} + 5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.3

Simplify.

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 \cdot (x + 9) + 5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.4

Apply the distributive property.

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 x + 2 \cdot 9 + 5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.5

Multiply $2$ by $9$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 x + 18 + 5 (- x - 18))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.6

Apply the distributive property.

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 x + 18 + 5 (- x) + 5 \cdot - 18)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.7

Multiply $- 1$ by $5$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 x + 18 - 5 x + 5 \cdot - 18)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.8

Multiply $5$ by $- 18$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (2 x + 18 - 5 x - 90)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.9

Subtract $5 x$ from $2 x$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (- 3 x + 18 - 90)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.10

Subtract $90$ from $18$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (- 3 x - 72)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.11

Factor $3$ out of $- 3 x - 72$ .

Tap for more steps...

Step 4.14.3.14.11.1

Factor $3$ out of $- 3 x$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (3 (- x) - 72)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.11.2

Factor $3$ out of $- 72$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (3 (- x) + 3 (- 24))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.14.11.3

Factor $3$ out of $3 (- x) + 3 (- 24)$ .

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} (3 (- x - 24))}{2}}{8 {(x + 9)}^{5}}$

$- \frac{3 \frac{{(x + 9)}^{\frac{3}{2}} \cdot 3 (- x - 24)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.3.15

Move $3$ to the left of ${(x + 9)}^{\frac{3}{2}}$ .

$- \frac{3 \frac{3 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.4

Combine terms.

Tap for more steps...

Step 4.14.4.1

Combine $3$ and $\frac{3 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2}$ .

$- \frac{\frac{3 (3 {(x + 9)}^{\frac{3}{2}} (- x - 24))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.4.2

Multiply $3$ by $3$ .

$- \frac{\frac{9 ({(x + 9)}^{\frac{3}{2}} (- x - 24))}{2}}{8 {(x + 9)}^{5}}$

Step 4.14.4.3

Rewrite $\frac{\frac{9 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2}}{8 {(x + 9)}^{5}}$ as a product.

$- (\frac{9 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2} \cdot \frac{1}{8 {(x + 9)}^{5}})$

Step 4.14.4.4

Multiply $\frac{9 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2}$ by $\frac{1}{8 {(x + 9)}^{5}}$ .

$- \frac{9 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{2 (8 {(x + 9)}^{5})}$

Step 4.14.4.5

Multiply $8$ by $2$ .

$- \frac{9 {(x + 9)}^{\frac{3}{2}} (- x - 24)}{16 {(x + 9)}^{5}}$

Step 4.14.4.6

Move ${(x + 9)}^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{5} {(x + 9)}^{- \frac{3}{2}}}$

Step 4.14.4.7

Multiply ${(x + 9)}^{5}$ by ${(x + 9)}^{- \frac{3}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.14.4.7.1

Move ${(x + 9)}^{- \frac{3}{2}}$ .

$- \frac{9 (- x - 24)}{16 ({(x + 9)}^{- \frac{3}{2}} {(x + 9)}^{5})}$

Step 4.14.4.7.2

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

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{- \frac{3}{2} + 5}}$

Step 4.14.4.7.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{- \frac{3}{2} + 5 \cdot \frac{2}{2}}}$

Step 4.14.4.7.4

Combine $5$ and $\frac{2}{2}$ .

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{- \frac{3}{2} + \frac{5 \cdot 2}{2}}}$

Step 4.14.4.7.5

Combine the numerators over the common denominator.

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{\frac{- 3 + 5 \cdot 2}{2}}}$

Step 4.14.4.7.6

Simplify the numerator.

Tap for more steps...

Step 4.14.4.7.6.1

Multiply $5$ by $2$ .

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{\frac{- 3 + 10}{2}}}$

Step 4.14.4.7.6.2

Add $- 3$ and $10$ .

$- \frac{9 (- x - 24)}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.5

Factor $- 1$ out of $- x$ .

$- \frac{9 (- (x) - 24)}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.6

Rewrite $- 24$ as $- 1 (24)$ .

$- \frac{9 (- (x) - 1 (24))}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.7

Factor $- 1$ out of $- (x) - 1 (24)$ .

$- \frac{9 (- (x + 24))}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.8

Rewrite $- (x + 24)$ as $- 1 (x + 24)$ .

$- \frac{9 (- 1 (x + 24))}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.9

Move the negative in front of the fraction.

$- - \frac{9 (x + 24)}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.10

Multiply $- 1$ by $- 1$ .

$1 \frac{9 (x + 24)}{16 {(x + 9)}^{\frac{7}{2}}}$

Step 4.14.11

Multiply $\frac{9 (x + 24)}{16 {(x + 9)}^{\frac{7}{2}}}$ by $1$ .

$f^{4} (x) = \frac{9 (x + 24)}{16 {(x + 9)}^{\frac{7}{2}}}$