Calculus Examples

$y = \frac{x^{2} + 4 x + 3}{\sqrt{x}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 4 x + 3$ and $g (x) = \sqrt{x}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $1$ .

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

Step 4

Differentiate.

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Step 4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{\sqrt{x} (2 x + 4 + 0) - (x^{2} + 4 x + 3) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $2 x + 4$ and $0$ .

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Simplify.

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Step 10.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Apply the distributive property.

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

Step 11.4

Simplify each term.

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Step 11.4.1

Rewrite using the commutative property of multiplication.

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

Step 11.4.2

Move $4$ to the left of $\sqrt{x}$ .

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

Step 11.4.3

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

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Step 11.4.3.1

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

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

Step 11.4.3.2

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

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

Step 11.4.3.3

Cancel the common factor.

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

Step 11.4.3.4

Rewrite the expression.

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

Step 11.4.4

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

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

Step 11.4.5

Multiply $4$ by $- 1$ .

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 4 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.6

Cancel the common factor of $2$ .

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Step 11.4.6.1

Factor $2$ out of $- 4 x$ .

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

Step 11.4.6.2

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

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

Step 11.4.6.3

Cancel the common factor.

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

Step 11.4.6.4

Rewrite the expression.

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x \frac{1}{x^{\frac{1}{2}}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.7

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x \frac{- 2}{x^{\frac{1}{2}}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.8

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + \frac{x \cdot - 2}{x^{\frac{1}{2}}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.9

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x \cdot - 2 x^{- \frac{1}{2}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10

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

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Step 11.4.10.1

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{- \frac{1}{2}} x \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10.2

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

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Step 11.4.10.2.1

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{- \frac{1}{2}} x^{1} \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10.2.2

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{- \frac{1}{2} + 1} \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10.3

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{- \frac{1}{2} + \frac{2}{2}} \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10.4

Combine the numerators over the common denominator.

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{\frac{- 1 + 2}{2}} \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.10.5

Add $- 1$ and $2$ .

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} + x^{\frac{1}{2}} \cdot - 2 - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.11

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 \cdot x^{\frac{1}{2}} - 1 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.12

Multiply $- 1$ by $3$ .

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - 3 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.13

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

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} + \frac{- 3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4.14

Move the negative in front of the fraction.

$\frac{2 \sqrt{x} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5

Combine terms.

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Step 11.5.1

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

$\frac{2 x^{\frac{1}{2}} x + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.2

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

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

Step 11.5.3

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

$\frac{2 x^{1 + \frac{1}{2}} + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.4

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

$\frac{2 x^{\frac{2}{2} + \frac{1}{2}} + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.5

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{2 + 1}{2}} + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.6

Add $2$ and $1$ .

$\frac{2 x^{\frac{3}{2}} + 4 \sqrt{x} - \frac{x^{\frac{3}{2}}}{2} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.7

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

$\frac{2 x^{\frac{3}{2}} \cdot \frac{2}{2} - \frac{x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.8

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

$\frac{\frac{2 x^{\frac{3}{2}} \cdot 2}{2} - \frac{x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.9

Combine the numerators over the common denominator.

$\frac{\frac{2 x^{\frac{3}{2}} \cdot 2 - x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.10

Multiply $2$ by $2$ .

$\frac{\frac{4 x^{\frac{3}{2}} - x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.11

Subtract $x^{\frac{3}{2}}$ from $4 x^{\frac{3}{2}}$ .

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.5.12

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 11.5.12.1

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

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

Step 11.5.12.2

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

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x^{\frac{1}{2} \cdot 2}}$

Step 11.5.12.3

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

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x^{\frac{2}{2}}}$

Step 11.5.12.4

Cancel the common factor of $2$ .

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Step 11.5.12.4.1

Cancel the common factor.

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x^{\frac{2}{2}}}$

Step 11.5.12.4.2

Rewrite the expression.

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x^{1}}$

Step 11.5.12.5

Simplify.

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x}$

Step 11.5.13

Multiply $\frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{3 x^{\frac{3}{2}}}{2} + 4 \sqrt{x} - 2 x^{\frac{1}{2}} - \frac{3}{2 x^{\frac{1}{2}}}}{x}$

Step 11.5.14

Combine.

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

Step 11.5.15

Apply the distributive property.

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

Step 11.5.16

Cancel the common factor of $2$ .

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Step 11.5.16.1

Cancel the common factor.

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

Step 11.5.16.2

Rewrite the expression.

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

Step 11.5.17

Multiply $4$ by $2$ .

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

Step 11.5.18

Multiply $- 2$ by $2$ .

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

Step 11.5.19

Multiply $- 1$ by $2$ .

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} - 2 \frac{3}{2 x^{\frac{1}{2}}}}{2 x}$

Step 11.5.20

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

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} + \frac{- 2 \cdot 3}{2 x^{\frac{1}{2}}}}{2 x}$

Step 11.5.21

Multiply $- 2$ by $3$ .

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} + \frac{- 6}{2 x^{\frac{1}{2}}}}{2 x}$

Step 11.5.22

Factor $2$ out of $- 6$ .

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} + \frac{2 \cdot - 3}{2 x^{\frac{1}{2}}}}{2 x}$

Step 11.5.23

Cancel the common factors.

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Step 11.5.23.1

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

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

Step 11.5.23.2

Cancel the common factor.

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} + \frac{2 \cdot - 3}{2 x^{\frac{1}{2}}}}{2 x}$

Step 11.5.23.3

Rewrite the expression.

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} + \frac{- 3}{x^{\frac{1}{2}}}}{2 x}$

Step 11.5.24

Move the negative in front of the fraction.

$\frac{3 x^{\frac{3}{2}} + 8 \sqrt{x} - 4 x^{\frac{1}{2}} - \frac{3}{x^{\frac{1}{2}}}}{2 x}$

Step 11.6

Reorder terms.

$\frac{3 x^{\frac{3}{2}} - 4 x^{\frac{1}{2}} - \frac{3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7

Simplify the numerator.

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Step 11.7.1

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

$\frac{- 4 x^{\frac{1}{2}} + \frac{3 x^{\frac{3}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - \frac{3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.2

Combine the numerators over the common denominator.

$\frac{- 4 x^{\frac{1}{2}} + \frac{3 x^{\frac{3}{2}} x^{\frac{1}{2}} - 3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.3

Simplify the numerator.

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Step 11.7.3.1

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

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Step 11.7.3.1.1

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

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

Step 11.7.3.1.2

Factor $3$ out of $- 3$ .

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

Step 11.7.3.1.3

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

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

Step 11.7.3.2

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

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Step 11.7.3.2.1

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

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

Step 11.7.3.2.2

Combine the numerators over the common denominator.

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

Step 11.7.3.2.3

Add $3$ and $1$ .

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

Step 11.7.3.2.4

Divide $4$ by $2$ .

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

Step 11.7.3.3

Rewrite $1$ as $1^{2}$ .

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

Step 11.7.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 11.7.4

To write $- 4 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 11.7.5

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

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

Step 11.7.6

Combine the numerators over the common denominator.

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

Step 11.7.7

Simplify the numerator.

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Step 11.7.7.1

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

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Step 11.7.7.1.1

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

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

Step 11.7.7.1.2

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

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

Step 11.7.7.1.3

Combine the numerators over the common denominator.

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

Step 11.7.7.1.4

Add $1$ and $1$ .

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

Step 11.7.7.1.5

Divide $2$ by $2$ .

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

Step 11.7.7.2

Simplify $- 4 x^{1}$ .

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

Step 11.7.7.3

Apply the distributive property.

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

Step 11.7.7.4

Multiply $3$ by $1$ .

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

Step 11.7.7.5

Expand $(3 x + 3) (x - 1)$ using the FOIL Method.

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Step 11.7.7.5.1

Apply the distributive property.

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

Step 11.7.7.5.2

Apply the distributive property.

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

Step 11.7.7.5.3

Apply the distributive property.

$\frac{\frac{- 4 x + 3 x \cdot x + 3 x \cdot - 1 + 3 x + 3 \cdot - 1}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.6

Simplify and combine like terms.

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Step 11.7.7.6.1

Simplify each term.

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Step 11.7.7.6.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 11.7.7.6.1.1.1

Move $x$ .

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

Step 11.7.7.6.1.1.2

Multiply $x$ by $x$ .

$\frac{\frac{- 4 x + 3 x^{2} + 3 x \cdot - 1 + 3 x + 3 \cdot - 1}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.6.1.2

Multiply $- 1$ by $3$ .

$\frac{\frac{- 4 x + 3 x^{2} - 3 x + 3 x + 3 \cdot - 1}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.6.1.3

Multiply $3$ by $- 1$ .

$\frac{\frac{- 4 x + 3 x^{2} - 3 x + 3 x - 3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.6.2

Add $- 3 x$ and $3 x$ .

$\frac{\frac{- 4 x + 3 x^{2} + 0 - 3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.6.3

Add $3 x^{2}$ and $0$ .

$\frac{\frac{- 4 x + 3 x^{2} - 3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.7.7

Reorder terms.

$\frac{\frac{3 x^{2} - 4 x - 3}{x^{\frac{1}{2}}} + 8 \sqrt{x}}{2 x}$

Step 11.7.8

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

$\frac{\frac{3 x^{2} - 4 x - 3}{x^{\frac{1}{2}}} + \frac{8 \sqrt{x} x^{\frac{1}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.9

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 \sqrt{x} x^{\frac{1}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10

Simplify the numerator.

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Step 11.7.10.1

Multiply $8 \sqrt{x} x^{\frac{1}{2}}$ .

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Step 11.7.10.1.1

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

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

Step 11.7.10.1.2

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

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x^{\frac{1}{2} + \frac{1}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.1.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x^{\frac{1 + 1}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.1.4

Add $1$ and $1$ .

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x^{\frac{2}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.1.5

Cancel the common factor of $2$ .

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Step 11.7.10.1.5.1

Cancel the common factor.

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x^{\frac{2}{2}}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.1.5.2

Rewrite the expression.

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x^{1}}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.2

Simplify.

$\frac{\frac{3 x^{2} - 4 x - 3 + 8 x}{x^{\frac{1}{2}}}}{2 x}$

Step 11.7.10.3

Add $- 4 x$ and $8 x$ .

$\frac{\frac{3 x^{2} + 4 x - 3}{x^{\frac{1}{2}}}}{2 x}$

Step 11.8

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x^{2} + 4 x - 3}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$

Step 11.9

Multiply $\frac{3 x^{2} + 4 x - 3}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$ .

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Step 11.9.1

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

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

Step 11.9.2

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

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Step 11.9.2.1

Move $x$ .

$\frac{3 x^{2} + 4 x - 3}{x \cdot x^{\frac{1}{2}} \cdot 2}$

Step 11.9.2.2

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

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Step 11.9.2.2.1

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

$\frac{3 x^{2} + 4 x - 3}{x^{1} x^{\frac{1}{2}} \cdot 2}$

Step 11.9.2.2.2

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

$\frac{3 x^{2} + 4 x - 3}{x^{1 + \frac{1}{2}} \cdot 2}$

Step 11.9.2.3

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

$\frac{3 x^{2} + 4 x - 3}{x^{\frac{2}{2} + \frac{1}{2}} \cdot 2}$

Step 11.9.2.4

Combine the numerators over the common denominator.

$\frac{3 x^{2} + 4 x - 3}{x^{\frac{2 + 1}{2}} \cdot 2}$

Step 11.9.2.5

Add $2$ and $1$ .

$\frac{3 x^{2} + 4 x - 3}{x^{\frac{3}{2}} \cdot 2}$

Step 11.10

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

$\frac{3 x^{2} + 4 x - 3}{2 x^{\frac{3}{2}}}$