Calculus Examples

$f (x) = \frac{x^{2} + 11}{\sqrt{x}}$

Step 1

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

$\frac{d}{d x} [\frac{x^{2} + 11}{x^{\frac{1}{2}}}]$

Step 2

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 $f (x) = x^{2} + 11$ and $g (x) = x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [x^{2} + 11] - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 3

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

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Step 3.1

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

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [x^{2} + 11] - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [x^{2} + 11] - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [x^{2} + 11] - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{1}}$

Step 4

Simplify.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [x^{2} + 11] - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 5

By the Sum Rule, the derivative of $x^{2} + 11$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [11]$ .

$\frac{x^{\frac{1}{2}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [11]) - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 6

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

$\frac{x^{\frac{1}{2}} (2 x + \frac{d}{d x} [11]) - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 7

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$\frac{x^{\frac{1}{2}} (2 x + 0) - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 8

Add $2 x$ and $0$ .

$\frac{x^{\frac{1}{2}} (2 x) - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9

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

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Step 9.1

Move $x$ .

$\frac{x \cdot x^{\frac{1}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9.2

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

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Step 9.2.1

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

$\frac{x^{1} x^{\frac{1}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9.2.2

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

$\frac{x^{1 + \frac{1}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9.3

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

$\frac{x^{\frac{2}{2} + \frac{1}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{x^{\frac{2 + 1}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 9.5

Add $2$ and $1$ .

$\frac{x^{\frac{3}{2}} \cdot 2 - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 10

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

$\frac{2 \cdot x^{\frac{3}{2}} - (x^{2} + 11) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 11

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

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{1}{2} - 1})}{x}$

Step 12

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

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{x}$

Step 13

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

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{x}$

Step 14

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{x}$

Step 15

Simplify the numerator.

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Step 15.1

Multiply $- 1$ by $2$ .

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{1 - 2}{2}})}{x}$

Step 15.2

Subtract $2$ from $1$ .

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{\frac{- 1}{2}})}{x}$

Step 16

Move the negative in front of the fraction.

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) (\frac{1}{2} x^{- \frac{1}{2}})}{x}$

Step 17

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

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) \frac{x^{- \frac{1}{2}}}{2}}{x}$

Step 18

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

$\frac{2 x^{\frac{3}{2}} - (x^{2} + 11) \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$\frac{2 x^{\frac{3}{2}} + (- x^{2} - 1 \cdot 11) \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.2

Apply the distributive property.

$\frac{2 x^{\frac{3}{2}} - x^{2} \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3

Simplify the numerator.

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Step 19.3.1

Simplify each term.

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Step 19.3.1.1

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

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Step 19.3.1.1.1

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

$\frac{2 x^{\frac{3}{2}} + x^{\frac{1}{2}} (- x^{\frac{3}{2}}) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.1.2

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

$\frac{2 x^{\frac{3}{2}} + x^{\frac{1}{2}} (- x^{\frac{3}{2}}) \frac{1}{x^{\frac{1}{2}} \cdot 2} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.1.3

Cancel the common factor.

$\frac{2 x^{\frac{3}{2}} + x^{\frac{1}{2}} (- x^{\frac{3}{2}}) \frac{1}{x^{\frac{1}{2}} \cdot 2} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.1.4

Rewrite the expression.

$\frac{2 x^{\frac{3}{2}} - x^{\frac{3}{2}} \frac{1}{2} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.2

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

$\frac{2 x^{\frac{3}{2}} - \frac{x^{\frac{3}{2}}}{2} - 1 \cdot 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.3

Multiply $- 1$ by $11$ .

$\frac{2 x^{\frac{3}{2}} - \frac{x^{\frac{3}{2}}}{2} - 11 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.4

Combine $- 11$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{2 x^{\frac{3}{2}} - \frac{x^{\frac{3}{2}}}{2} + \frac{- 11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.1.5

Move the negative in front of the fraction.

$\frac{2 x^{\frac{3}{2}} - \frac{x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.2

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

$\frac{2 x^{\frac{3}{2}} \cdot \frac{2}{2} - \frac{x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.3

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

$\frac{\frac{2 x^{\frac{3}{2}} \cdot 2}{2} - \frac{x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.4

Combine the numerators over the common denominator.

$\frac{\frac{2 x^{\frac{3}{2}} \cdot 2 - x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5

Simplify each term.

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Step 19.3.5.1

Simplify the numerator.

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Step 19.3.5.1.1

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

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Step 19.3.5.1.1.1

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

$\frac{\frac{2 \cdot 2 x^{\frac{3}{2}} - x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.1.1.2

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

$\frac{\frac{x^{\frac{3}{2}} (2 \cdot 2) - x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.1.1.3

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

$\frac{\frac{x^{\frac{3}{2}} (2 \cdot 2) + x^{\frac{3}{2}} \cdot - 1}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.1.1.4

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

$\frac{\frac{x^{\frac{3}{2}} (2 \cdot 2 - 1)}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.1.2

Multiply $2$ by $2$ .

$\frac{\frac{x^{\frac{3}{2}} (4 - 1)}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.1.3

Subtract $1$ from $4$ .

$\frac{\frac{x^{\frac{3}{2}} \cdot 3}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.3.5.2

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

$\frac{\frac{3 x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.4

Combine terms.

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Step 19.4.1

Multiply $\frac{\frac{3 x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{3 x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}}}{x}$

Step 19.4.2

Combine.

$\frac{2 (\frac{3 x^{\frac{3}{2}}}{2} - \frac{11}{2 x^{\frac{1}{2}}})}{2 x}$

Step 19.4.3

Apply the distributive property.

$\frac{2 \frac{3 x^{\frac{3}{2}}}{2} + 2 (- \frac{11}{2 x^{\frac{1}{2}}})}{2 x}$

Step 19.4.4

Cancel the common factor of $2$ .

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Step 19.4.4.1

Cancel the common factor.

$\frac{2 \frac{3 x^{\frac{3}{2}}}{2} + 2 (- \frac{11}{2 x^{\frac{1}{2}}})}{2 x}$

Step 19.4.4.2

Rewrite the expression.

$\frac{3 x^{\frac{3}{2}} + 2 (- \frac{11}{2 x^{\frac{1}{2}}})}{2 x}$

Step 19.4.5

Multiply $- 1$ by $2$ .

$\frac{3 x^{\frac{3}{2}} - 2 \frac{11}{2 x^{\frac{1}{2}}}}{2 x}$

Step 19.4.6

Combine $- 2$ and $\frac{11}{2 x^{\frac{1}{2}}}$ .

$\frac{3 x^{\frac{3}{2}} + \frac{- 2 \cdot 11}{2 x^{\frac{1}{2}}}}{2 x}$

Step 19.4.7

Multiply $- 2$ by $11$ .

$\frac{3 x^{\frac{3}{2}} + \frac{- 22}{2 x^{\frac{1}{2}}}}{2 x}$

Step 19.4.8

Factor $2$ out of $- 22$ .

$\frac{3 x^{\frac{3}{2}} + \frac{2 \cdot - 11}{2 x^{\frac{1}{2}}}}{2 x}$

Step 19.4.9

Cancel the common factors.

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Step 19.4.9.1

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

$\frac{3 x^{\frac{3}{2}} + \frac{2 \cdot - 11}{2 (x^{\frac{1}{2}})}}{2 x}$

Step 19.4.9.2

Cancel the common factor.

$\frac{3 x^{\frac{3}{2}} + \frac{2 \cdot - 11}{2 x^{\frac{1}{2}}}}{2 x}$

Step 19.4.9.3

Rewrite the expression.

$\frac{3 x^{\frac{3}{2}} + \frac{- 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.4.10

Move the negative in front of the fraction.

$\frac{3 x^{\frac{3}{2}} - \frac{11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5

Simplify the numerator.

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Step 19.5.1

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

$\frac{\frac{3 x^{\frac{3}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - \frac{11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.2

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{\frac{3}{2}} x^{\frac{1}{2}} - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.3

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

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Step 19.5.3.1

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

$\frac{\frac{3 (x^{\frac{1}{2}} x^{\frac{3}{2}}) - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.3.2

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

$\frac{\frac{3 x^{\frac{1}{2} + \frac{3}{2}} - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.3.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{\frac{1 + 3}{2}} - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.3.4

Add $1$ and $3$ .

$\frac{\frac{3 x^{\frac{4}{2}} - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.5.3.5

Divide $4$ by $2$ .

$\frac{\frac{3 x^{2} - 11}{x^{\frac{1}{2}}}}{2 x}$

Step 19.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x^{2} - 11}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$

Step 19.7

Multiply $\frac{3 x^{2} - 11}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$ .

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Step 19.7.1

Multiply $\frac{3 x^{2} - 11}{x^{\frac{1}{2}}}$ by $\frac{1}{2 x}$ .

$\frac{3 x^{2} - 11}{x^{\frac{1}{2}} (2 x)}$

Step 19.7.2

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

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Step 19.7.2.1

Move $x$ .

$\frac{3 x^{2} - 11}{x \cdot x^{\frac{1}{2}} \cdot 2}$

Step 19.7.2.2

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

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Step 19.7.2.2.1

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

$\frac{3 x^{2} - 11}{x^{1} x^{\frac{1}{2}} \cdot 2}$

Step 19.7.2.2.2

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

$\frac{3 x^{2} - 11}{x^{1 + \frac{1}{2}} \cdot 2}$

Step 19.7.2.3

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

$\frac{3 x^{2} - 11}{x^{\frac{2}{2} + \frac{1}{2}} \cdot 2}$

Step 19.7.2.4

Combine the numerators over the common denominator.

$\frac{3 x^{2} - 11}{x^{\frac{2 + 1}{2}} \cdot 2}$

Step 19.7.2.5

Add $2$ and $1$ .

$\frac{3 x^{2} - 11}{x^{\frac{3}{2}} \cdot 2}$

Step 19.8

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

$\frac{3 x^{2} - 11}{2 x^{\frac{3}{2}}}$