Calculus Examples

$f (x) = 6 x^{5} + 4 x^{\frac{1}{3}} \sqrt{x} - 10 e^{\frac{x}{2}} + 3^{x}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $6 x^{5} + 4 x^{\frac{1}{3}} \sqrt{x} - 10 e^{\frac{x}{2}} + 3^{x}$ with respect to $x$ is $\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$ .

$\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x^{5}]$ .

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.2.2

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

$6 (5 x^{4}) + \frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.2.3

Multiply $5$ by $6$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [4 x^{\frac{1}{3}} \sqrt{x}]$ .

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Step 1.3.1

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

$30 x^{4} + \frac{d}{d x} [4 (x^{\frac{1}{2}} x^{\frac{1}{3}})] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.2

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

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{1}{2} + \frac{1}{3}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.3

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

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.4

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

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{2}{2}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{3}{2 \cdot 3} + \frac{1}{3} \cdot \frac{2}{2}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.5.2

Multiply $2$ by $3$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{3}{6} + \frac{1}{3} \cdot \frac{2}{2}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.5.3

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{3}{6} + \frac{2}{3 \cdot 2}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.5.4

Multiply $3$ by $2$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{3}{6} + \frac{2}{6}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.6

Combine the numerators over the common denominator.

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{3 + 2}{6}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.7

Add $3$ and $2$ .

$30 x^{4} + \frac{d}{d x} [4 x^{\frac{5}{6}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.8

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{5}{6}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{5}{6}}]$ .

$30 x^{4} + 4 \frac{d}{d x} [x^{\frac{5}{6}}] + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{6}$ .

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{5}{6} - 1}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{5}{6} - 1 \cdot \frac{6}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.11

Combine $- 1$ and $\frac{6}{6}$ .

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{5}{6} + \frac{- 1 \cdot 6}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.12

Combine the numerators over the common denominator.

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{5 - 1 \cdot 6}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.13

Simplify the numerator.

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Step 1.3.13.1

Multiply $- 1$ by $6$ .

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{5 - 6}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.13.2

Subtract $6$ from $5$ .

$30 x^{4} + 4 (\frac{5}{6} x^{\frac{- 1}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.14

Move the negative in front of the fraction.

$30 x^{4} + 4 (\frac{5}{6} x^{- \frac{1}{6}}) + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.15

Combine $\frac{5}{6}$ and $x^{- \frac{1}{6}}$ .

$30 x^{4} + 4 \frac{5 x^{- \frac{1}{6}}}{6} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.16

Combine $4$ and $\frac{5 x^{- \frac{1}{6}}}{6}$ .

$30 x^{4} + \frac{4 (5 x^{- \frac{1}{6}})}{6} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.17

Multiply $5$ by $4$ .

$30 x^{4} + \frac{20 x^{- \frac{1}{6}}}{6} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.18

Move $x^{- \frac{1}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$30 x^{4} + \frac{20}{6 x^{\frac{1}{6}}} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.19

Factor $2$ out of $20$ .

$30 x^{4} + \frac{2 (10)}{6 x^{\frac{1}{6}}} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.20

Cancel the common factors.

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Step 1.3.20.1

Factor $2$ out of $6 x^{\frac{1}{6}}$ .

$30 x^{4} + \frac{2 (10)}{2 (3 x^{\frac{1}{6}})} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.20.2

Cancel the common factor.

$30 x^{4} + \frac{2 \cdot 10}{2 (3 x^{\frac{1}{6}})} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.3.20.3

Rewrite the expression.

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{d}{d x} [- 10 e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.4

Evaluate $\frac{d}{d x} [- 10 e^{\frac{x}{2}}]$ .

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Step 1.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 e^{\frac{x}{2}}$ with respect to $x$ is $- 10 \frac{d}{d x} [e^{\frac{x}{2}}]$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 \frac{d}{d x} [e^{\frac{x}{2}}] + \frac{d}{d x} [3^{x}]$

Step 1.4.2

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

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Step 1.4.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [3^{x}]$

Step 1.4.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (e^{u} \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [3^{x}]$

Step 1.4.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [3^{x}]$

Step 1.4.3

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

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x])) + \frac{d}{d x} [3^{x}]$

Step 1.4.4

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

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (e^{\frac{x}{2}} (\frac{1}{2} \cdot 1)) + \frac{d}{d x} [3^{x}]$

Step 1.4.5

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

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 (e^{\frac{x}{2}} \frac{1}{2}) + \frac{d}{d x} [3^{x}]$

Step 1.4.6

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

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 10 \frac{e^{\frac{x}{2}}}{2} + \frac{d}{d x} [3^{x}]$

Step 1.4.7

Combine $- 10$ and $\frac{e^{\frac{x}{2}}}{2}$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{- 10 e^{\frac{x}{2}}}{2} + \frac{d}{d x} [3^{x}]$

Step 1.4.8

Cancel the common factor of $- 10$ and $2$ .

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Step 1.4.8.1

Factor $2$ out of $- 10 e^{\frac{x}{2}}$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{2 (- 5 e^{\frac{x}{2}})}{2} + \frac{d}{d x} [3^{x}]$

Step 1.4.8.2

Cancel the common factors.

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Step 1.4.8.2.1

Factor $2$ out of $2$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{2 (- 5 e^{\frac{x}{2}})}{2 (1)} + \frac{d}{d x} [3^{x}]$

Step 1.4.8.2.2

Cancel the common factor.

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{2 (- 5 e^{\frac{x}{2}})}{2 \cdot 1} + \frac{d}{d x} [3^{x}]$

Step 1.4.8.2.3

Rewrite the expression.

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} + \frac{- 5 e^{\frac{x}{2}}}{1} + \frac{d}{d x} [3^{x}]$

Step 1.4.8.2.4

Divide $- 5 e^{\frac{x}{2}}$ by $1$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 5 e^{\frac{x}{2}} + \frac{d}{d x} [3^{x}]$

Step 1.5

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$30 x^{4} + \frac{10}{3 x^{\frac{1}{6}}} - 5 e^{\frac{x}{2}} + 3^{x} \ln (3)$

Step 1.6

Reorder terms.

$f' (x) = 30 x^{4} + 3^{x} \ln (3) + \frac{10}{3 x^{\frac{1}{6}}} - 5 e^{\frac{x}{2}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $30 x^{4} + 3^{x} \ln (3) + \frac{10}{3 x^{\frac{1}{6}}} - 5 e^{\frac{x}{2}}$ with respect to $x$ is $\frac{d}{d x} [30 x^{4}] + \frac{d}{d x} [3^{x} \ln (3)] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$ .

$\frac{d}{d x} [30 x^{4}] + \frac{d}{d x} [3^{x} \ln (3)] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.2

Evaluate $\frac{d}{d x} [30 x^{4}]$ .

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Step 2.2.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{4}$ with respect to $x$ is $30 \frac{d}{d x} [x^{4}]$ .

$30 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3^{x} \ln (3)] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$30 (4 x^{3}) + \frac{d}{d x} [3^{x} \ln (3)] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.2.3

Multiply $4$ by $30$ .

$120 x^{3} + \frac{d}{d x} [3^{x} \ln (3)] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3

Evaluate $\frac{d}{d x} [3^{x} \ln (3)]$ .

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Step 2.3.1

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

$120 x^{3} + \ln (3) \frac{d}{d x} [3^{x}] + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$120 x^{3} + \ln (3) (3^{x} \ln (3)) + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3.3

Raise $\ln (3)$ to the power of $1$ .

$120 x^{3} + 3^{x} (\ln^{1} (3) \ln (3)) + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3.4

Raise $\ln (3)$ to the power of $1$ .

$120 x^{3} + 3^{x} (\ln^{1} (3) \ln^{1} (3)) + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3.5

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

$120 x^{3} + 3^{x} {\ln (3)}^{1 + 1} + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.3.6

Add $1$ and $1$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4

Evaluate $\frac{d}{d x} [\frac{10}{3 x^{\frac{1}{6}}}]$ .

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Step 2.4.1

Since $\frac{10}{3}$ is constant with respect to $x$ , the derivative of $\frac{10}{3 x^{\frac{1}{6}}}$ with respect to $x$ is $\frac{10}{3} \frac{d}{d x} [\frac{1}{x^{\frac{1}{6}}}]$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} \frac{d}{d x} [\frac{1}{x^{\frac{1}{6}}}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.2

Rewrite $\frac{1}{x^{\frac{1}{6}}}$ as ${(x^{\frac{1}{6}})}^{- 1}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} \frac{d}{d x} [{(x^{\frac{1}{6}})}^{- 1}] + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{6}}$ .

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Step 2.4.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{6}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{6}}]) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.3.2

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

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{1}{6}}]) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{6}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- {(x^{\frac{1}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{6}}]) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{6}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- {(x^{\frac{1}{6}})}^{- 2} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5

Multiply the exponents in ${(x^{\frac{1}{6}})}^{- 2}$ .

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Step 2.4.5.1

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

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{1}{6} \cdot - 2} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.2

Cancel the common factor of $2$ .

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Step 2.4.5.2.1

Factor $2$ out of $6$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{1}{2 (3)} \cdot - 2} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.2.2

Factor $2$ out of $- 2$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{1}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.2.3

Cancel the common factor.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{1}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.2.4

Rewrite the expression.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{1}{3} \cdot - 1} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.3

Combine $\frac{1}{3}$ and $- 1$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{\frac{- 1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.5.4

Move the negative in front of the fraction.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1 \cdot \frac{6}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.7

Combine $- 1$ and $\frac{6}{6}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} + \frac{- 1 \cdot 6}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.8

Combine the numerators over the common denominator.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1 - 1 \cdot 6}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.9

Simplify the numerator.

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Step 2.4.9.1

Multiply $- 1$ by $6$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1 - 6}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.9.2

Subtract $6$ from $1$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{- 5}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.10

Move the negative in front of the fraction.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} (\frac{1}{6} x^{- \frac{5}{6}})) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.11

Combine $\frac{1}{6}$ and $x^{- \frac{5}{6}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- x^{- \frac{1}{3}} \frac{x^{- \frac{5}{6}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.12

Combine $\frac{x^{- \frac{5}{6}}}{6}$ and $x^{- \frac{1}{3}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{5}{6}} x^{- \frac{1}{3}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13

Multiply $x^{- \frac{5}{6}}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 2.4.13.1

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

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{5}{6} - \frac{1}{3}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{5}{6} - \frac{1}{3} \cdot \frac{2}{2}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.13.3.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{5}{6} - \frac{2}{3 \cdot 2}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.3.2

Multiply $3$ by $2$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{5}{6} - \frac{2}{6}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.4

Combine the numerators over the common denominator.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{\frac{- 5 - 2}{6}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.5

Subtract $2$ from $- 5$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{\frac{- 7}{6}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.13.6

Move the negative in front of the fraction.

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{x^{- \frac{7}{6}}}{6}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.14

Move $x^{- \frac{7}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) + \frac{10}{3} (- \frac{1}{6 x^{\frac{7}{6}}}) + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.15

Multiply $\frac{10}{3}$ by $\frac{1}{6 x^{\frac{7}{6}}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{10}{3 (6 x^{\frac{7}{6}})} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.16

Multiply $6$ by $3$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{10}{18 x^{\frac{7}{6}}} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.17

Factor $2$ out of $10$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{2 (5)}{18 x^{\frac{7}{6}}} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.18

Cancel the common factors.

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Step 2.4.18.1

Factor $2$ out of $18 x^{\frac{7}{6}}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{2 (5)}{2 (9 x^{\frac{7}{6}})} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.18.2

Cancel the common factor.

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{2 \cdot 5}{2 (9 x^{\frac{7}{6}})} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.4.18.3

Rewrite the expression.

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} + \frac{d}{d x} [- 5 e^{\frac{x}{2}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- 5 e^{\frac{x}{2}}]$ .

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Step 2.5.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 e^{\frac{x}{2}}$ with respect to $x$ is $- 5 \frac{d}{d x} [e^{\frac{x}{2}}]$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 \frac{d}{d x} [e^{\frac{x}{2}}]$

Step 2.5.2

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

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Step 2.5.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [\frac{x}{2}])$

Step 2.5.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (e^{u_{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 2.5.2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 2.5.3

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

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x]))$

Step 2.5.4

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

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (e^{\frac{x}{2}} (\frac{1}{2} \cdot 1))$

Step 2.5.5

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

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 (e^{\frac{x}{2}} \frac{1}{2})$

Step 2.5.6

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

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - 5 \frac{e^{\frac{x}{2}}}{2}$

Step 2.5.7

Combine $- 5$ and $\frac{e^{\frac{x}{2}}}{2}$ .

$120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} + \frac{- 5 e^{\frac{x}{2}}}{2}$

Step 2.5.8

Move the negative in front of the fraction.

$f'' (x) = 120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - \frac{5 e^{\frac{x}{2}}}{2}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $120 x^{3} + 3^{x} \ln^{2} (3) - \frac{5}{9 x^{\frac{7}{6}}} - \frac{5 e^{\frac{x}{2}}}{2}$ with respect to $x$ is $\frac{d}{d x} [120 x^{3}] + \frac{d}{d x} [3^{x} \ln^{2} (3)] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$ .

$\frac{d}{d x} [120 x^{3}] + \frac{d}{d x} [3^{x} \ln^{2} (3)] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [120 x^{3}]$ .

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Step 3.2.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x^{3}$ with respect to $x$ is $120 \frac{d}{d x} [x^{3}]$ .

$120 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3^{x} \ln^{2} (3)] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.2.2

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

$120 (3 x^{2}) + \frac{d}{d x} [3^{x} \ln^{2} (3)] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.2.3

Multiply $3$ by $120$ .

$360 x^{2} + \frac{d}{d x} [3^{x} \ln^{2} (3)] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3

Evaluate $\frac{d}{d x} [3^{x} \ln^{2} (3)]$ .

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Step 3.3.1

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

$360 x^{2} + \ln^{2} (3) \frac{d}{d x} [3^{x}] + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$360 x^{2} + \ln^{2} (3) (3^{x} \ln (3)) + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3.3

Multiply $\ln^{2} (3)$ by $\ln (3)$ by adding the exponents.

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Step 3.3.3.1

Move $\ln (3)$ .

$360 x^{2} + \ln (3) \ln^{2} (3) \cdot 3^{x} + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3.3.2

Multiply $\ln (3)$ by $\ln^{2} (3)$ .

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Step 3.3.3.2.1

Raise $\ln (3)$ to the power of $1$ .

$360 x^{2} + \ln^{1} (3) \ln^{2} (3) \cdot 3^{x} + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3.3.2.2

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

$360 x^{2} + {\ln (3)}^{1 + 2} \cdot 3^{x} + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.3.3.3

Add $1$ and $2$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{5}{9 x^{\frac{7}{6}}}]$ .

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Step 3.4.1

Since $- \frac{5}{9}$ is constant with respect to $x$ , the derivative of $- \frac{5}{9 x^{\frac{7}{6}}}$ with respect to $x$ is $- \frac{5}{9} \frac{d}{d x} [\frac{1}{x^{\frac{7}{6}}}]$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} \frac{d}{d x} [\frac{1}{x^{\frac{7}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.2

Rewrite $\frac{1}{x^{\frac{7}{6}}}$ as ${(x^{\frac{7}{6}})}^{- 1}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} \frac{d}{d x} [{(x^{\frac{7}{6}})}^{- 1}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{7}{6}}$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{7}{6}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{7}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.3.2

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{7}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{7}{6}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- {(x^{\frac{7}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{6}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- {(x^{\frac{7}{6}})}^{- 2} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5

Multiply the exponents in ${(x^{\frac{7}{6}})}^{- 2}$ .

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Step 3.4.5.1

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7}{6} \cdot - 2} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.2

Cancel the common factor of $2$ .

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Step 3.4.5.2.1

Factor $2$ out of $6$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7}{2 (3)} \cdot - 2} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.2.2

Factor $2$ out of $- 2$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.2.3

Cancel the common factor.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.2.4

Rewrite the expression.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7}{3} \cdot - 1} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.3

Combine $\frac{7}{3}$ and $- 1$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{7 \cdot - 1}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.4

Multiply $7$ by $- 1$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{\frac{- 7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.5.5

Move the negative in front of the fraction.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1 \cdot \frac{6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.7

Combine $- 1$ and $\frac{6}{6}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} + \frac{- 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.8

Combine the numerators over the common denominator.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7 - 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $6$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7 - 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.9.2

Subtract $6$ from $7$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{1}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.10

Combine $\frac{7}{6}$ and $x^{\frac{1}{6}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- x^{- \frac{7}{3}} \frac{7 x^{\frac{1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.11

Combine $\frac{7 x^{\frac{1}{6}}}{6}$ and $x^{- \frac{7}{3}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{\frac{1}{6}} x^{- \frac{7}{3}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12

Multiply $x^{\frac{1}{6}}$ by $x^{- \frac{7}{3}}$ by adding the exponents.

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Step 3.4.12.1

Move $x^{- \frac{7}{3}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 (x^{- \frac{7}{3}} x^{\frac{1}{6}})}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.2

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{- \frac{7}{3} + \frac{1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.3

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{- \frac{7}{3} \cdot \frac{2}{2} + \frac{1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.12.4.1

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{- \frac{7 \cdot 2}{3 \cdot 2} + \frac{1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.4.2

Multiply $3$ by $2$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{- \frac{7 \cdot 2}{6} + \frac{1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.5

Combine the numerators over the common denominator.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{\frac{- 7 \cdot 2 + 1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.6

Simplify the numerator.

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Step 3.4.12.6.1

Multiply $- 7$ by $2$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{\frac{- 14 + 1}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.6.2

Add $- 14$ and $1$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{\frac{- 13}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.12.7

Move the negative in front of the fraction.

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7 x^{- \frac{13}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.13

Move $x^{- \frac{13}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} - \frac{5}{9} (- \frac{7}{6 x^{\frac{13}{6}}}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.14

Multiply $- 1$ by $- 1$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + 1 (\frac{5}{9}) \frac{7}{6 x^{\frac{13}{6}}} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.15

Multiply $\frac{5}{9}$ by $1$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{5}{9} \cdot \frac{7}{6 x^{\frac{13}{6}}} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.16

Multiply $\frac{5}{9}$ by $\frac{7}{6 x^{\frac{13}{6}}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{5 \cdot 7}{9 (6 x^{\frac{13}{6}})} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.17

Multiply $5$ by $7$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{9 (6 x^{\frac{13}{6}})} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.4.18

Multiply $6$ by $9$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$

Step 3.5

Evaluate $\frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{2}]$ .

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Step 3.5.1

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5 e^{\frac{x}{2}}}{2}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [e^{\frac{x}{2}}]$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} \frac{d}{d x} [e^{\frac{x}{2}}]$

Step 3.5.2

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

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Step 3.5.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [\frac{x}{2}])$

Step 3.5.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (e^{u_{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 3.5.2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 3.5.3

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x]))$

Step 3.5.4

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (e^{\frac{x}{2}} (\frac{1}{2} \cdot 1))$

Step 3.5.5

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} (e^{\frac{x}{2}} \frac{1}{2})$

Step 3.5.6

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

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5}{2} \cdot \frac{e^{\frac{x}{2}}}{2}$

Step 3.5.7

Multiply $\frac{e^{\frac{x}{2}}}{2}$ by $\frac{5}{2}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{e^{\frac{x}{2}} \cdot 5}{2 \cdot 2}$

Step 3.5.8

Multiply $2$ by $2$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{e^{\frac{x}{2}} \cdot 5}{4}$

Step 3.5.9

Move $5$ to the left of $e^{\frac{x}{2}}$ .

$360 x^{2} + \ln^{3} (3) \cdot 3^{x} + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5 e^{\frac{x}{2}}}{4}$

Step 3.6

Reorder terms.

$f''' (x) = 360 x^{2} + 3^{x} \ln^{3} (3) + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5 e^{\frac{x}{2}}}{4}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $360 x^{2} + 3^{x} \ln^{3} (3) + \frac{35}{54 x^{\frac{13}{6}}} - \frac{5 e^{\frac{x}{2}}}{4}$ with respect to $x$ is $\frac{d}{d x} [360 x^{2}] + \frac{d}{d x} [3^{x} \ln^{3} (3)] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$ .

$\frac{d}{d x} [360 x^{2}] + \frac{d}{d x} [3^{x} \ln^{3} (3)] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.2

Evaluate $\frac{d}{d x} [360 x^{2}]$ .

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Step 4.2.1

Since $360$ is constant with respect to $x$ , the derivative of $360 x^{2}$ with respect to $x$ is $360 \frac{d}{d x} [x^{2}]$ .

$360 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3^{x} \ln^{3} (3)] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.2.2

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

$360 (2 x) + \frac{d}{d x} [3^{x} \ln^{3} (3)] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.2.3

Multiply $2$ by $360$ .

$720 x + \frac{d}{d x} [3^{x} \ln^{3} (3)] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3

Evaluate $\frac{d}{d x} [3^{x} \ln^{3} (3)]$ .

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Step 4.3.1

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

$720 x + \ln^{3} (3) \frac{d}{d x} [3^{x}] + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$720 x + \ln^{3} (3) (3^{x} \ln (3)) + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3.3

Multiply $\ln^{3} (3)$ by $\ln (3)$ by adding the exponents.

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Step 4.3.3.1

Move $\ln (3)$ .

$720 x + \ln (3) \ln^{3} (3) \cdot 3^{x} + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3.3.2

Multiply $\ln (3)$ by $\ln^{3} (3)$ .

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Step 4.3.3.2.1

Raise $\ln (3)$ to the power of $1$ .

$720 x + \ln^{1} (3) \ln^{3} (3) \cdot 3^{x} + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3.3.2.2

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

$720 x + {\ln (3)}^{1 + 3} \cdot 3^{x} + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.3.3.3

Add $1$ and $3$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4

Evaluate $\frac{d}{d x} [\frac{35}{54 x^{\frac{13}{6}}}]$ .

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Step 4.4.1

Since $\frac{35}{54}$ is constant with respect to $x$ , the derivative of $\frac{35}{54 x^{\frac{13}{6}}}$ with respect to $x$ is $\frac{35}{54} \frac{d}{d x} [\frac{1}{x^{\frac{13}{6}}}]$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} \frac{d}{d x} [\frac{1}{x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.2

Rewrite $\frac{1}{x^{\frac{13}{6}}}$ as ${(x^{\frac{13}{6}})}^{- 1}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} \frac{d}{d x} [{(x^{\frac{13}{6}})}^{- 1}] + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{13}{6}}$ .

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Step 4.4.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{13}{6}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.3.2

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

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{13}{6}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- {(x^{\frac{13}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{13}{6}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- {(x^{\frac{13}{6}})}^{- 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5

Multiply the exponents in ${(x^{\frac{13}{6}})}^{- 2}$ .

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Step 4.4.5.1

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

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13}{6} \cdot - 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.2

Cancel the common factor of $2$ .

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Step 4.4.5.2.1

Factor $2$ out of $6$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13}{2 (3)} \cdot - 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.2.2

Factor $2$ out of $- 2$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.2.3

Cancel the common factor.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.2.4

Rewrite the expression.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13}{3} \cdot - 1} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.3

Combine $\frac{13}{3}$ and $- 1$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{13 \cdot - 1}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.4

Multiply $13$ by $- 1$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{\frac{- 13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.5.5

Move the negative in front of the fraction.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1 \cdot \frac{6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.7

Combine $- 1$ and $\frac{6}{6}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} + \frac{- 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.8

Combine the numerators over the common denominator.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13 - 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.9

Simplify the numerator.

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Step 4.4.9.1

Multiply $- 1$ by $6$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13 - 6}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.9.2

Subtract $6$ from $13$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{7}{6}})) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.10

Combine $\frac{13}{6}$ and $x^{\frac{7}{6}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- x^{- \frac{13}{3}} \frac{13 x^{\frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.11

Combine $\frac{13 x^{\frac{7}{6}}}{6}$ and $x^{- \frac{13}{3}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{\frac{7}{6}} x^{- \frac{13}{3}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12

Multiply $x^{\frac{7}{6}}$ by $x^{- \frac{13}{3}}$ by adding the exponents.

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Step 4.4.12.1

Move $x^{- \frac{13}{3}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 (x^{- \frac{13}{3}} x^{\frac{7}{6}})}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.2

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

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{- \frac{13}{3} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.3

To write $- \frac{13}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{- \frac{13}{3} \cdot \frac{2}{2} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.12.4.1

Multiply $\frac{13}{3}$ by $\frac{2}{2}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{- \frac{13 \cdot 2}{3 \cdot 2} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.4.2

Multiply $3$ by $2$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{- \frac{13 \cdot 2}{6} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.5

Combine the numerators over the common denominator.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{\frac{- 13 \cdot 2 + 7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.6

Simplify the numerator.

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Step 4.4.12.6.1

Multiply $- 13$ by $2$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{\frac{- 26 + 7}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.6.2

Add $- 26$ and $7$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{\frac{- 19}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.12.7

Move the negative in front of the fraction.

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13 x^{- \frac{19}{6}}}{6}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.13

Move $x^{- \frac{19}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} + \frac{35}{54} (- \frac{13}{6 x^{\frac{19}{6}}}) + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.14

Multiply $\frac{35}{54}$ by $\frac{13}{6 x^{\frac{19}{6}}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{35 \cdot 13}{54 (6 x^{\frac{19}{6}})} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.15

Multiply $35$ by $13$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{54 (6 x^{\frac{19}{6}})} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.4.16

Multiply $6$ by $54$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} + \frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$

Step 4.5

Evaluate $\frac{d}{d x} [- \frac{5 e^{\frac{x}{2}}}{4}]$ .

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Step 4.5.1

Since $- \frac{5}{4}$ is constant with respect to $x$ , the derivative of $- \frac{5 e^{\frac{x}{2}}}{4}$ with respect to $x$ is $- \frac{5}{4} \frac{d}{d x} [e^{\frac{x}{2}}]$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} \frac{d}{d x} [e^{\frac{x}{2}}]$

Step 4.5.2

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

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Step 4.5.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [\frac{x}{2}])$

Step 4.5.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (e^{u_{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 4.5.2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}])$

Step 4.5.3

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

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x]))$

Step 4.5.4

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

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (e^{\frac{x}{2}} (\frac{1}{2} \cdot 1))$

Step 4.5.5

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

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} (e^{\frac{x}{2}} \frac{1}{2})$

Step 4.5.6

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

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5}{4} \cdot \frac{e^{\frac{x}{2}}}{2}$

Step 4.5.7

Multiply $\frac{e^{\frac{x}{2}}}{2}$ by $\frac{5}{4}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{e^{\frac{x}{2}} \cdot 5}{2 \cdot 4}$

Step 4.5.8

Multiply $2$ by $4$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{e^{\frac{x}{2}} \cdot 5}{8}$

Step 4.5.9

Move $5$ to the left of $e^{\frac{x}{2}}$ .

$720 x + \ln^{4} (3) \cdot 3^{x} - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5 e^{\frac{x}{2}}}{8}$

Step 4.6

Reorder terms.

$f^{4} (x) = 3^{x} \ln^{4} (3) + 720 x - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5 e^{\frac{x}{2}}}{8}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $3^{x} \ln^{4} (3) + 720 x - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5 e^{\frac{x}{2}}}{8}$ .

$3^{x} \ln^{4} (3) + 720 x - \frac{455}{324 x^{\frac{19}{6}}} - \frac{5 e^{\frac{x}{2}}}{8}$