Calculus Examples

$f (x) = (4 x + 2 \sqrt{x} - 1) (4 \sqrt{x} + 5)$

Step 1

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 $f (x) = 4 x + 2 \sqrt{x} - 1$ and $g (x) = 4 \sqrt{x} + 5$ .

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

Step 2

By the Sum Rule, the derivative of $4 \sqrt{x} + 5$ with respect to $x$ is $\frac{d}{d x} [4 \sqrt{x}] + \frac{d}{d x} [5]$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

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

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

Step 3.10

Combine $4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 3.11

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

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

Step 3.12

Factor $2$ out of $4$ .

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

Step 3.13

Cancel the common factors.

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Step 3.13.1

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

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

Step 3.13.2

Cancel the common factor.

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

Step 3.13.3

Rewrite the expression.

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Add $\frac{2}{x^{\frac{1}{2}}}$ and $0$ .

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

Step 4.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply $4$ by $1$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Combine the numerators over the common denominator.

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

Step 6.7

Simplify the numerator.

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Step 6.7.1

Multiply $- 1$ by $2$ .

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

Step 6.7.2

Subtract $2$ from $1$ .

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

Step 6.8

Move the negative in front of the fraction.

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

Step 6.9

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

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

Step 6.10

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

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

Step 6.11

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

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

Step 6.12

Cancel the common factor.

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

Step 6.13

Rewrite the expression.

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

Step 7

Differentiate using the Constant Rule.

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Step 7.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 7.2

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

$(4 x + 2 \sqrt{x} - 1) \frac{2}{x^{\frac{1}{2}}} + (4 \sqrt{x} + 5) (4 + \frac{1}{x^{\frac{1}{2}}})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$4 x \frac{2}{x^{\frac{1}{2}}} + 2 \sqrt{x} \frac{2}{x^{\frac{1}{2}}} - 1 \frac{2}{x^{\frac{1}{2}}} + (4 \sqrt{x} + 5) (4 + \frac{1}{x^{\frac{1}{2}}})$

Step 8.2

Combine terms.

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Step 8.2.1

Combine $4$ and $\frac{2}{x^{\frac{1}{2}}}$ .

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

Step 8.2.2

Multiply $4$ by $2$ .

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

Step 8.2.3

Combine $x$ and $\frac{8}{x^{\frac{1}{2}}}$ .

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

Step 8.2.4

Move $8$ to the left of $x$ .

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

Step 8.2.5

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

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

Step 8.2.6

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

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Step 8.2.6.1

Move $x^{- \frac{1}{2}}$ .

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

Step 8.2.6.2

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

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Step 8.2.6.2.1

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

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

Step 8.2.6.2.2

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

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

Step 8.2.6.3

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

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

Step 8.2.6.4

Combine the numerators over the common denominator.

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

Step 8.2.6.5

Add $- 1$ and $2$ .

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

Step 8.2.7

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

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

Step 8.2.8

Multiply $2$ by $2$ .

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

Step 8.2.9

Combine $\frac{4}{x^{\frac{1}{2}}}$ and $\sqrt{x}$ .

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

Step 8.2.10

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

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

Step 8.2.11

To write $(4 \sqrt{x} + 5) (4 + \frac{1}{x^{\frac{1}{2}}})$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 8.2.12

Combine the numerators over the common denominator.

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

Step 8.2.13

To write $8 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 8.2.14

Combine the numerators over the common denominator.

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

Step 8.2.15

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

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Step 8.2.15.1

Move $x^{\frac{1}{2}}$ .

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

Step 8.2.15.2

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

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

Step 8.2.15.3

Combine the numerators over the common denominator.

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

Step 8.2.15.4

Add $1$ and $1$ .

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

Step 8.2.15.5

Divide $2$ by $2$ .

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

Step 8.2.16

Simplify $8 x^{1}$ .

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

Step 8.2.17

Combine the numerators over the common denominator.

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

Step 8.3

Reorder terms.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Apply the distributive property.

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

Step 8.4.2

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

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

Step 8.4.3

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

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Step 8.4.3.1

Cancel the common factor.

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

Step 8.4.3.2

Rewrite the expression.

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

Step 8.4.4

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

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Step 8.4.4.1

Apply the distributive property.

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

Step 8.4.4.2

Apply the distributive property.

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

Step 8.4.4.3

Apply the distributive property.

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

Step 8.4.5

Simplify each term.

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Step 8.4.5.1

Multiply $4 x^{\frac{1}{2}} (4 \sqrt{x})$ .

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Step 8.4.5.1.1

Multiply $4$ by $4$ .

$\frac{16 x^{\frac{1}{2}} \sqrt{x} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.2

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

$\frac{16 (x^{\frac{1}{2}} x^{\frac{1}{2}}) + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.3

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

$\frac{16 x^{\frac{1}{2} + \frac{1}{2}} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.4

Combine the numerators over the common denominator.

$\frac{16 x^{\frac{1 + 1}{2}} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.5

Add $1$ and $1$ .

$\frac{16 x^{\frac{2}{2}} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.6

Cancel the common factor of $2$ .

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Step 8.4.5.1.6.1

Cancel the common factor.

$\frac{16 x^{\frac{2}{2}} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.1.6.2

Rewrite the expression.

$\frac{16 x^{1} + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.2

Simplify.

$\frac{16 x + 4 x^{\frac{1}{2}} \cdot 5 + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.3

Multiply $5$ by $4$ .

$\frac{16 x + 20 x^{\frac{1}{2}} + 1 (4 \sqrt{x}) + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.4

Multiply $4 \sqrt{x}$ by $1$ .

$\frac{16 x + 20 x^{\frac{1}{2}} + 4 \sqrt{x} + 1 \cdot 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.5.5

Multiply $5$ by $1$ .

$\frac{16 x + 20 x^{\frac{1}{2}} + 4 \sqrt{x} + 5 + 8 x - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.6

Add $16 x$ and $8 x$ .

$\frac{24 x + 20 x^{\frac{1}{2}} + 4 \sqrt{x} + 5 - 2 + 4 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.7

Add $4 \sqrt{x}$ and $4 \sqrt{x}$ .

$\frac{24 x + 20 x^{\frac{1}{2}} + 5 - 2 + 8 \sqrt{x}}{x^{\frac{1}{2}}}$

Step 8.4.8

Subtract $2$ from $5$ .

$\frac{24 x + 20 x^{\frac{1}{2}} + 3 + 8 \sqrt{x}}{x^{\frac{1}{2}}}$