Calculus Examples

$G (x) = (3 x^{2} + 5) (4 x + \sqrt{x})$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [(3 x^{2} + 5) (4 x + x^{\frac{1}{2}})]$

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 $f (x) = 3 x^{2} + 5$ and $g (x) = 4 x + x^{\frac{1}{2}}$ .

$(3 x^{2} + 5) \frac{d}{d x} [4 x + x^{\frac{1}{2}}] + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $4 x + x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$(3 x^{2} + 5) (\frac{d}{d x} [4 x] + \frac{d}{d x} [x^{\frac{1}{2}}]) + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.3.2

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

$(3 x^{2} + 5) (4 \frac{d}{d x} [x] + \frac{d}{d x} [x^{\frac{1}{2}}]) + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.3.3

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

$(3 x^{2} + 5) (4 \cdot 1 + \frac{d}{d x} [x^{\frac{1}{2}}]) + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.3.4

Multiply $4$ by $1$ .

$(3 x^{2} + 5) (4 + \frac{d}{d x} [x^{\frac{1}{2}}]) + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.3.5

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) \frac{d}{d x} [3 x^{2} + 5]$

Step 1.9

By the Sum Rule, the derivative of $3 x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5]$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5])$

Step 1.10

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])$

Step 1.11

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) (3 (2 x) + \frac{d}{d x} [5])$

Step 1.12

Multiply $2$ by $3$ .

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

Step 1.13

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) (6 x + 0)$

Step 1.14

Simplify the expression.

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Step 1.14.1

Add $6 x$ and $0$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (4 x + x^{\frac{1}{2}}) (6 x)$

Step 1.14.2

Move $6$ to the left of $4 x + x^{\frac{1}{2}}$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 6 \cdot (4 x + x^{\frac{1}{2}}) x$

Step 1.15

Simplify.

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Step 1.15.1

Apply the distributive property.

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + (6 (4 x) + 6 x^{\frac{1}{2}}) x$

Step 1.15.2

Apply the distributive property.

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 6 (4 x) x + 6 x^{\frac{1}{2}} x$

Step 1.15.3

Combine terms.

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Step 1.15.3.1

Multiply $4$ by $6$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x \cdot x + 6 x^{\frac{1}{2}} x$

Step 1.15.3.2

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 (x^{1} x) + 6 x^{\frac{1}{2}} x$

Step 1.15.3.3

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 (x^{1} x^{1}) + 6 x^{\frac{1}{2}} x$

Step 1.15.3.4

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{1 + 1} + 6 x^{\frac{1}{2}} x$

Step 1.15.3.5

Add $1$ and $1$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 x^{\frac{1}{2}} x$

Step 1.15.3.6

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 (x^{1} x^{\frac{1}{2}})$

Step 1.15.3.7

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 x^{1 + \frac{1}{2}}$

Step 1.15.3.8

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

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 x^{\frac{2}{2} + \frac{1}{2}}$

Step 1.15.3.9

Combine the numerators over the common denominator.

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 x^{\frac{2 + 1}{2}}$

Step 1.15.3.10

Add $2$ and $1$ .

$(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 24 x^{2} + 6 x^{\frac{3}{2}}$

Step 1.15.4

Reorder terms.

$24 x^{2} + (3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 6 x^{\frac{3}{2}}$

Step 1.15.5

Simplify each term.

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Step 1.15.5.1

Expand $(3 x^{2} + 5) (4 + \frac{1}{2 x^{\frac{1}{2}}})$ using the FOIL Method.

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Step 1.15.5.1.1

Apply the distributive property.

$24 x^{2} + 3 x^{2} (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 5 (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 6 x^{\frac{3}{2}}$

Step 1.15.5.1.2

Apply the distributive property.

$24 x^{2} + 3 x^{2} \cdot 4 + 3 x^{2} \frac{1}{2 x^{\frac{1}{2}}} + 5 (4 + \frac{1}{2 x^{\frac{1}{2}}}) + 6 x^{\frac{3}{2}}$

Step 1.15.5.1.3

Apply the distributive property.

$24 x^{2} + 3 x^{2} \cdot 4 + 3 x^{2} \frac{1}{2 x^{\frac{1}{2}}} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2

Simplify each term.

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Step 1.15.5.2.1

Multiply $4$ by $3$ .

$24 x^{2} + 12 x^{2} + 3 x^{2} \frac{1}{2 x^{\frac{1}{2}}} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.2

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 1.15.5.2.2.1

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

$24 x^{2} + 12 x^{2} + x^{\frac{1}{2}} (3 x^{\frac{3}{2}}) \frac{1}{2 x^{\frac{1}{2}}} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.2.2

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

$24 x^{2} + 12 x^{2} + x^{\frac{1}{2}} (3 x^{\frac{3}{2}}) \frac{1}{x^{\frac{1}{2}} \cdot 2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.2.3

Cancel the common factor.

$24 x^{2} + 12 x^{2} + x^{\frac{1}{2}} (3 x^{\frac{3}{2}}) \frac{1}{x^{\frac{1}{2}} \cdot 2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.2.4

Rewrite the expression.

$24 x^{2} + 12 x^{2} + 3 x^{\frac{3}{2}} \frac{1}{2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.3

Combine $3$ and $\frac{1}{2}$ .

$24 x^{2} + 12 x^{2} + x^{\frac{3}{2}} \frac{3}{2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.4

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

$24 x^{2} + 12 x^{2} + \frac{x^{\frac{3}{2}} \cdot 3}{2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.5

Move $3$ to the left of $x^{\frac{3}{2}}$ .

$24 x^{2} + 12 x^{2} + \frac{3 \cdot x^{\frac{3}{2}}}{2} + 5 \cdot 4 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.6

Multiply $5$ by $4$ .

$24 x^{2} + 12 x^{2} + \frac{3 x^{\frac{3}{2}}}{2} + 20 + 5 \frac{1}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.5.2.7

Combine $5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$24 x^{2} + 12 x^{2} + \frac{3 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.6

Add $24 x^{2}$ and $12 x^{2}$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}} + 6 x^{\frac{3}{2}}$

Step 1.15.7

To write $6 x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}}}{2} + 6 x^{\frac{3}{2}} \cdot \frac{2}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.8

Combine $6 x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}}}{2} + \frac{6 x^{\frac{3}{2}} \cdot 2}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.9

Combine the numerators over the common denominator.

$36 x^{2} + \frac{3 x^{\frac{3}{2}} + 6 x^{\frac{3}{2}} \cdot 2}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10

Simplify the numerator.

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Step 1.15.10.1

Factor $3 x^{\frac{3}{2}}$ out of $3 x^{\frac{3}{2}} + 6 x^{\frac{3}{2}} \cdot 2$ .

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Step 1.15.10.1.1

Move $x^{\frac{3}{2}}$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}} + 6 \cdot 2 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.1.2

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

$36 x^{2} + \frac{3 x^{\frac{3}{2}} (1) + 6 \cdot 2 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.1.3

Factor $3 x^{\frac{3}{2}}$ out of $6 \cdot 2 x^{\frac{3}{2}}$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}} (1) + 3 x^{\frac{3}{2}} (2 \cdot 2)}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.1.4

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

$36 x^{2} + \frac{3 x^{\frac{3}{2}} (1 + 2 \cdot 2)}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.2

Multiply $2$ by $2$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}} (1 + 4)}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.3

Add $1$ and $4$ .

$36 x^{2} + \frac{3 x^{\frac{3}{2}} \cdot 5}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 1.15.10.4

Multiply $5$ by $3$ .

$f' (x) = 36 x^{2} + \frac{15 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $36 x^{2} + \frac{15 x^{\frac{3}{2}}}{2} + 20 + \frac{5}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.2

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 2.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.2.2

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

$36 (2 x) + \frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.2.3

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{15 x^{\frac{3}{2}}}{2}]$ .

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Step 2.3.1

Since $\frac{15}{2}$ is constant with respect to $x$ , the derivative of $\frac{15 x^{\frac{3}{2}}}{2}$ with respect to $x$ is $\frac{15}{2} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$72 x + \frac{15}{2} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.2

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

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.3

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

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.4

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

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.5

Combine the numerators over the common denominator.

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $2$ .

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.6.2

Subtract $2$ from $3$ .

$72 x + \frac{15}{2} (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.7

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

$72 x + \frac{15}{2} \cdot \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.8

Multiply $\frac{15}{2}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

$72 x + \frac{15 (3 x^{\frac{1}{2}})}{2 \cdot 2} + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.9

Multiply $3$ by $15$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{2 \cdot 2} + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.3.10

Multiply $2$ by $2$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + \frac{d}{d x} [20] + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.4

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$

Step 2.5

Evaluate $\frac{d}{d x} [\frac{5}{2 x^{\frac{1}{2}}}]$ .

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Step 2.5.1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 2.5.2

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 2.5.3

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

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Step 2.5.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.5.3.2

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.5.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.5.4

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.5.5

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

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Step 2.5.5.1

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.5.5.2

Cancel the common factor of $2$ .

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Step 2.5.5.2.1

Factor $2$ out of $- 2$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.5.5.2.2

Cancel the common factor.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.5.5.2.3

Rewrite the expression.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.5.6

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 2.5.7

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 2.5.8

Combine the numerators over the common denominator.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 2.5.9

Simplify the numerator.

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Step 2.5.9.1

Multiply $- 1$ by $2$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 2.5.9.2

Subtract $2$ from $1$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 2.5.10

Move the negative in front of the fraction.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 2.5.11

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 2.5.12

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 2.5.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.5.13.1

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 2.5.13.2

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 2.5.13.3

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 2.5.13.4

Combine the numerators over the common denominator.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 2.5.13.5

Simplify the numerator.

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Step 2.5.13.5.1

Multiply $- 1$ by $2$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 2.5.13.5.2

Subtract $2$ from $- 1$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 2.5.13.6

Move the negative in front of the fraction.

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{x^{- \frac{3}{2}}}{2})$

Step 2.5.14

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

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 + \frac{5}{2} (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 2.5.15

Multiply $\frac{5}{2}$ by $\frac{1}{2 x^{\frac{3}{2}}}$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 - \frac{5}{2 (2 x^{\frac{3}{2}})}$

Step 2.5.16

Multiply $2$ by $2$ .

$72 x + \frac{45 x^{\frac{1}{2}}}{4} + 0 - \frac{5}{4 x^{\frac{3}{2}}}$

Step 2.6

Add $72 x + \frac{45 x^{\frac{1}{2}}}{4}$ and $0$ .

$f'' (x) = 72 x + \frac{45 x^{\frac{1}{2}}}{4} - \frac{5}{4 x^{\frac{3}{2}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $72 x + \frac{45 x^{\frac{1}{2}}}{4} - \frac{5}{4 x^{\frac{3}{2}}}$ with respect to $x$ is $\frac{d}{d x} [72 x] + \frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$ .

$\frac{d}{d x} [72 x] + \frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.2

Evaluate $\frac{d}{d x} [72 x]$ .

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Step 3.2.1

Since $72$ is constant with respect to $x$ , the derivative of $72 x$ with respect to $x$ is $72 \frac{d}{d x} [x]$ .

$72 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.2.2

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

$72 \cdot 1 + \frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.2.3

Multiply $72$ by $1$ .

$72 + \frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{45 x^{\frac{1}{2}}}{4}]$ .

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Step 3.3.1

Since $\frac{45}{4}$ is constant with respect to $x$ , the derivative of $\frac{45 x^{\frac{1}{2}}}{4}$ with respect to $x$ is $\frac{45}{4} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$72 + \frac{45}{4} \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.2

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

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.3

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

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.4

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

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.5

Combine the numerators over the common denominator.

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.6

Simplify the numerator.

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Step 3.3.6.1

Multiply $- 1$ by $2$ .

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.6.2

Subtract $2$ from $1$ .

$72 + \frac{45}{4} (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.7

Move the negative in front of the fraction.

$72 + \frac{45}{4} (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.8

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

$72 + \frac{45}{4} \cdot \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.9

Multiply $\frac{45}{4}$ by $\frac{x^{- \frac{1}{2}}}{2}$ .

$72 + \frac{45 x^{- \frac{1}{2}}}{4 \cdot 2} + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.10

Multiply $4$ by $2$ .

$72 + \frac{45 x^{- \frac{1}{2}}}{8} + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.3.11

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{5}{4 x^{\frac{3}{2}}}]$ .

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Step 3.4.1

Since $- \frac{5}{4}$ is constant with respect to $x$ , the derivative of $- \frac{5}{4 x^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{5}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 3.4.2

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}]$

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

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Step 3.4.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{3}{2}}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{3}{2}}])$

Step 3.4.3.2

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- u^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}])$

Step 3.4.3.3

Replace all occurrences of $u$ with $x^{\frac{3}{2}}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- {(x^{\frac{3}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}])$

Step 3.4.4

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.4.5

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

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Step 3.4.5.1

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

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 $- 2$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.4.5.2.2

Cancel the common factor.

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.4.5.2.3

Rewrite the expression.

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.4.5.3

Multiply $3$ by $- 1$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 3.4.6

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.4.7

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.4.8

Combine the numerators over the common denominator.

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}))$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $2$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}}))$

Step 3.4.9.2

Subtract $2$ from $3$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{1}{2}}))$

Step 3.4.10

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- x^{- 3} \frac{3 x^{\frac{1}{2}}}{2})$

Step 3.4.11

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{\frac{1}{2}} x^{- 3}}{2})$

Step 3.4.12

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

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Step 3.4.12.1

Move $x^{- 3}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2})$

Step 3.4.12.2

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{- 3 + \frac{1}{2}}}{2})$

Step 3.4.12.3

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2})$

Step 3.4.12.4

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

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2})$

Step 3.4.12.5

Combine the numerators over the common denominator.

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{\frac{- 3 \cdot 2 + 1}{2}}}{2})$

Step 3.4.12.6

Simplify the numerator.

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Step 3.4.12.6.1

Multiply $- 3$ by $2$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{\frac{- 6 + 1}{2}}}{2})$

Step 3.4.12.6.2

Add $- 6$ and $1$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{\frac{- 5}{2}}}{2})$

Step 3.4.12.7

Move the negative in front of the fraction.

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3 x^{- \frac{5}{2}}}{2})$

Step 3.4.13

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} - \frac{5}{4} (- \frac{3}{2 x^{\frac{5}{2}}})$

Step 3.4.14

Multiply $- 1$ by $- 1$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} + 1 (\frac{5}{4}) \frac{3}{2 x^{\frac{5}{2}}}$

Step 3.4.15

Multiply $\frac{5}{4}$ by $1$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{5}{4} \cdot \frac{3}{2 x^{\frac{5}{2}}}$

Step 3.4.16

Multiply $\frac{5}{4}$ by $\frac{3}{2 x^{\frac{5}{2}}}$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{5 \cdot 3}{4 (2 x^{\frac{5}{2}})}$

Step 3.4.17

Multiply $5$ by $3$ .

$72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{15}{4 (2 x^{\frac{5}{2}})}$

Step 3.4.18

Multiply $2$ by $4$ .

$f''' (x) = 72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{15}{8 x^{\frac{5}{2}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $72 + \frac{45}{8 x^{\frac{1}{2}}} + \frac{15}{8 x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{d}{d x} [72] + \frac{d}{d x} [\frac{45}{8 x^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$ .

$\frac{d}{d x} [72] + \frac{d}{d x} [\frac{45}{8 x^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.1.2

Since $72$ is constant with respect to $x$ , the derivative of $72$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{45}{8 x^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2

Evaluate $\frac{d}{d x} [\frac{45}{8 x^{\frac{1}{2}}}]$ .

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Step 4.2.1

Since $\frac{45}{8}$ is constant with respect to $x$ , the derivative of $\frac{45}{8 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{45}{8} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$0 + \frac{45}{8} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.2

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

$0 + \frac{45}{8} \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}] + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.3

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

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}}$ .

$0 + \frac{45}{8} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.3.2

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

$0 + \frac{45}{8} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.3.3

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

$0 + \frac{45}{8} (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.4

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

$0 + \frac{45}{8} (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.5

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

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Step 4.2.5.1

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

$0 + \frac{45}{8} (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.5.2

Cancel the common factor of $2$ .

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Step 4.2.5.2.1

Factor $2$ out of $- 2$ .

$0 + \frac{45}{8} (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.5.2.2

Cancel the common factor.

$0 + \frac{45}{8} (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.5.2.3

Rewrite the expression.

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.6

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

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.7

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

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.8

Combine the numerators over the common denominator.

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Multiply $- 1$ by $2$ .

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.9.2

Subtract $2$ from $1$ .

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.10

Move the negative in front of the fraction.

$0 + \frac{45}{8} (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.11

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

$0 + \frac{45}{8} (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.12

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

$0 + \frac{45}{8} (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.2.13.1

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

$0 + \frac{45}{8} (- \frac{x^{- \frac{1}{2} - 1}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.2

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

$0 + \frac{45}{8} (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.3

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

$0 + \frac{45}{8} (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.4

Combine the numerators over the common denominator.

$0 + \frac{45}{8} (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.5

Simplify the numerator.

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Step 4.2.13.5.1

Multiply $- 1$ by $2$ .

$0 + \frac{45}{8} (- \frac{x^{\frac{- 1 - 2}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.5.2

Subtract $2$ from $- 1$ .

$0 + \frac{45}{8} (- \frac{x^{\frac{- 3}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.13.6

Move the negative in front of the fraction.

$0 + \frac{45}{8} (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.14

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

$0 + \frac{45}{8} (- \frac{1}{2 x^{\frac{3}{2}}}) + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.15

Multiply $\frac{45}{8}$ by $\frac{1}{2 x^{\frac{3}{2}}}$ .

$0 - \frac{45}{8 (2 x^{\frac{3}{2}})} + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.2.16

Multiply $2$ by $8$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{15}{8 x^{\frac{5}{2}}}]$ .

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Step 4.3.1

Since $\frac{15}{8}$ is constant with respect to $x$ , the derivative of $\frac{15}{8 x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{15}{8} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$

Step 4.3.2

Rewrite $\frac{1}{x^{\frac{5}{2}}}$ as ${(x^{\frac{5}{2}})}^{- 1}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} \frac{d}{d x} [{(x^{\frac{5}{2}})}^{- 1}]$

Step 4.3.3

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 $g (x) = x^{\frac{5}{2}}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{5}{2}}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.3.3.2

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{5}{2}}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- {(x^{\frac{5}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.3.4

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- {(x^{\frac{5}{2}})}^{- 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.5

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

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Step 4.3.5.1

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{\frac{5}{2} \cdot - 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.5.2

Cancel the common factor of $2$ .

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Step 4.3.5.2.1

Factor $2$ out of $- 2$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{\frac{5}{2} \cdot (2 (- 1))} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.5.2.2

Cancel the common factor.

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{\frac{5}{2} \cdot (2 \cdot - 1)} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.5.2.3

Rewrite the expression.

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{5 \cdot - 1} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.5.3

Multiply $5$ by $- 1$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.3.6

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Step 4.3.7

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 4.3.8

Combine the numerators over the common denominator.

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}))$

Step 4.3.9

Simplify the numerator.

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Step 4.3.9.1

Multiply $- 1$ by $2$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 2}{2}}))$

Step 4.3.9.2

Subtract $2$ from $5$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{3}{2}}))$

Step 4.3.10

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- x^{- 5} \frac{5 x^{\frac{3}{2}}}{2})$

Step 4.3.11

Combine $\frac{5 x^{\frac{3}{2}}}{2}$ and $x^{- 5}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{\frac{3}{2}} x^{- 5}}{2})$

Step 4.3.12

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

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Step 4.3.12.1

Move $x^{- 5}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 (x^{- 5} x^{\frac{3}{2}})}{2})$

Step 4.3.12.2

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{- 5 + \frac{3}{2}}}{2})$

Step 4.3.12.3

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{- 5 \cdot \frac{2}{2} + \frac{3}{2}}}{2})$

Step 4.3.12.4

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

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{\frac{- 5 \cdot 2}{2} + \frac{3}{2}}}{2})$

Step 4.3.12.5

Combine the numerators over the common denominator.

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{\frac{- 5 \cdot 2 + 3}{2}}}{2})$

Step 4.3.12.6

Simplify the numerator.

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Step 4.3.12.6.1

Multiply $- 5$ by $2$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{\frac{- 10 + 3}{2}}}{2})$

Step 4.3.12.6.2

Add $- 10$ and $3$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{\frac{- 7}{2}}}{2})$

Step 4.3.12.7

Move the negative in front of the fraction.

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5 x^{- \frac{7}{2}}}{2})$

Step 4.3.13

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} + \frac{15}{8} (- \frac{5}{2 x^{\frac{7}{2}}})$

Step 4.3.14

Multiply $\frac{15}{8}$ by $\frac{5}{2 x^{\frac{7}{2}}}$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} - \frac{15 \cdot 5}{8 (2 x^{\frac{7}{2}})}$

Step 4.3.15

Multiply $15$ by $5$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} - \frac{75}{8 (2 x^{\frac{7}{2}})}$

Step 4.3.16

Multiply $2$ by $8$ .

$0 - \frac{45}{16 x^{\frac{3}{2}}} - \frac{75}{16 x^{\frac{7}{2}}}$

Step 4.4

Subtract $\frac{45}{16 x^{\frac{3}{2}}}$ from $0$ .

$f^{4} (x) = - \frac{45}{16 x^{\frac{3}{2}}} - \frac{75}{16 x^{\frac{7}{2}}}$

Step 5

The fourth derivative of $G (x)$ with respect to $x$ is $- \frac{45}{16 x^{\frac{3}{2}}} - \frac{75}{16 x^{\frac{7}{2}}}$ .

$- \frac{45}{16 x^{\frac{3}{2}}} - \frac{75}{16 x^{\frac{7}{2}}}$