Calculus Examples

$y = \frac{x^{2} - 2 \sqrt{x}}{x}$

Step 1

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

$\frac{d}{d x} [\frac{x^{2} - 2 x^{\frac{1}{2}}}{x}]$

Step 2

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

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Step 2.1

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

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} x^{\frac{3}{2}} - 2 x^{\frac{1}{2}}}{x}]$

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x \cdot x^{- \frac{1}{2}}}]$

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x^{1} x^{- \frac{1}{2}}}]$

Step 4.1.2

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

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x^{1 - \frac{1}{2}}}]$

Step 4.2

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

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x^{\frac{2}{2} - \frac{1}{2}}}]$

Step 4.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x^{\frac{2 - 1}{2}}}]$

Step 4.4

Subtract $1$ from $2$ .

$\frac{d}{d x} [\frac{x^{\frac{3}{2}} - 2}{x^{\frac{1}{2}}}]$

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

Simplify.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Subtract $2$ from $3$ .

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

Step 14

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

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

Step 15

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 16

Combine fractions.

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Step 16.1

Add $\frac{3 x^{\frac{1}{2}}}{2}$ and $0$ .

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

Step 16.2

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

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

Step 17

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

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

Combine the numerators over the common denominator.

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

Step 17.4

Add $1$ and $1$ .

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

Step 17.5

Divide $2$ by $2$ .

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

Step 18

Simplify $x^{1} \cdot 3$ .

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

Step 19

Move $3$ to the left of $x$ .

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

Step 20

Multiply $\frac{\frac{3 x}{2} - (x^{\frac{3}{2}} - 2) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$ by $\frac{2}{2}$ .

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

Step 21

Simplify terms.

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Step 21.1

Combine.

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

Step 21.2

Apply the distributive property.

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

Step 21.3

Cancel the common factor of $2$ .

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Step 21.3.1

Cancel the common factor.

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

Step 21.3.2

Rewrite the expression.

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

Step 21.4

Multiply $- 1$ by $2$ .

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

Step 22

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

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

Step 23

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

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

Step 24

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

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

Step 25

Combine the numerators over the common denominator.

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

Step 26

Simplify the numerator.

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Step 26.1

Multiply $- 1$ by $2$ .

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

Step 26.2

Subtract $2$ from $1$ .

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

Step 27

Move the negative in front of the fraction.

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

Step 28

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

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

Step 29

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

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

Step 30

Simplify the expression.

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Step 30.1

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

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

Step 30.2

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

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

Step 31

Factor $2$ out of $- 2$ .

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

Step 32

Cancel the common factors.

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Step 32.1

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

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

Step 32.2

Cancel the common factor.

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

Step 32.3

Rewrite the expression.

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

Step 33

Move the negative in front of the fraction.

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

Step 34

Simplify.

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Step 34.1

Apply the distributive property.

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

Step 34.2

Simplify the numerator.

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Step 34.2.1

Simplify each term.

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Step 34.2.1.1

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

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Step 34.2.1.1.1

Move the leading negative in $- \frac{1}{x^{\frac{1}{2}}}$ into the numerator.

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

Step 34.2.1.1.2

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

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

Step 34.2.1.1.3

Cancel the common factor.

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

Step 34.2.1.1.4

Rewrite the expression.

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

Step 34.2.1.2

Divide $2$ by $2$ .

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

Step 34.2.1.3

Simplify.

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

Step 34.2.1.4

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

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Step 34.2.1.4.1

Multiply $- 2$ by $- 1$ .

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

Step 34.2.1.4.2

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

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

Step 34.2.2

Subtract $x$ from $3 x$ .

$\frac{2 x + \frac{2}{x^{\frac{1}{2}}}}{2 x}$

Step 34.3

Combine terms.

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Step 34.3.1

Factor $2$ out of $2 x$ .

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

Step 34.3.2

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

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

Step 34.3.3

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

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

Step 34.3.4

Cancel the common factors.

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Step 34.3.4.1

Factor $2$ out of $2 x$ .

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

Step 34.3.4.2

Cancel the common factor.

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

Step 34.3.4.3

Rewrite the expression.

$\frac{x + \frac{1}{x^{\frac{1}{2}}}}{x}$

Step 34.3.5

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

$\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}} \cdot \frac{x + \frac{1}{x^{\frac{1}{2}}}}{x}$

Step 34.3.6

Combine.

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

Step 34.3.7

Apply the distributive property.

$\frac{x^{\frac{1}{2}} x + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{2}}}}{x^{\frac{1}{2}} x}$

Step 34.3.8

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

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Step 34.3.8.1

Cancel the common factor.

$\frac{x^{\frac{1}{2}} x + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{2}}}}{x^{\frac{1}{2}} x}$

Step 34.3.8.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} x + 1}{x^{\frac{1}{2}} x}$

Step 34.3.9

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

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Step 34.3.9.1

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

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Step 34.3.9.1.1

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

$\frac{x^{\frac{1}{2}} x^{1} + 1}{x^{\frac{1}{2}} x}$

Step 34.3.9.1.2

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

$\frac{x^{\frac{1}{2} + 1} + 1}{x^{\frac{1}{2}} x}$

Step 34.3.9.2

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

$\frac{x^{\frac{1}{2} + \frac{2}{2}} + 1}{x^{\frac{1}{2}} x}$

Step 34.3.9.3

Combine the numerators over the common denominator.

$\frac{x^{\frac{1 + 2}{2}} + 1}{x^{\frac{1}{2}} x}$

Step 34.3.9.4

Add $1$ and $2$ .

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

Step 34.3.10

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

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Step 34.3.10.1

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

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Step 34.3.10.1.1

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

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

Step 34.3.10.1.2

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

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

Step 34.3.10.2

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

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

Step 34.3.10.3

Combine the numerators over the common denominator.

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

Step 34.3.10.4

Add $1$ and $2$ .

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