Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{2}}}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Simplify the expression.

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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Step 8.3.3.1

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

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

Step 8.3.3.2

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

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

Step 8.3.3.3

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Subtract $2$ from $- 1$ .

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

Step 14

Move the negative in front of the fraction.

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

Step 15

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

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Apply the distributive property.

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

Step 17.2

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

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

Step 17.3

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

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Step 17.3.1

Apply the distributive property.

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

Step 17.3.2

Apply the distributive property.

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

Step 17.3.3

Apply the distributive property.

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

Step 17.4

Simplify and combine like terms.

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Step 17.4.1

Simplify each term.

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Step 17.4.1.1

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

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Step 17.4.1.1.1

Cancel the common factor.

Step 17.4.1.1.2

Rewrite the expression.

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

Step 17.4.1.2

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

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Step 17.4.1.2.1

Move the leading negative in $- \frac{1}{2 x^{\frac{3}{2}}}$ into the numerator.

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

Step 17.4.1.2.2

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

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

Step 17.4.1.2.3

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

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

Step 17.4.1.2.4

Cancel the common factor.

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

Step 17.4.1.2.5

Rewrite the expression.

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

Step 17.4.1.3

Cancel the common factor of $2$ .

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Step 17.4.1.3.1

Cancel the common factor.

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

Step 17.4.1.3.2

Rewrite the expression.

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

Step 17.4.1.4

Simplify.

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

Step 17.4.1.5

Move the negative in front of the fraction.

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

Step 17.4.1.6

Combine.

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

Step 17.4.1.7

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

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Step 17.4.1.7.1

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

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

Step 17.4.1.7.2

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

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

Step 17.4.1.7.3

Combine the numerators over the common denominator.

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

Step 17.4.1.7.4

Add $1$ and $1$ .

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

Step 17.4.1.7.5

Divide $2$ by $2$ .

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

Step 17.4.1.8

Simplify $x^{1} \cdot 2$ .

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

Step 17.4.1.9

Multiply $2$ by $1$ .

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

Step 17.4.1.10

Cancel the common factor of $2$ .

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Step 17.4.1.10.1

Cancel the common factor.

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

Step 17.4.1.10.2

Rewrite the expression.

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

Step 17.4.1.11

Rewrite using the commutative property of multiplication.

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

Step 17.4.1.12

Cancel the common factor of $2$ .

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Step 17.4.1.12.1

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

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

Step 17.4.1.12.2

Factor $2$ out of $- 2$ .

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

Step 17.4.1.12.3

Cancel the common factor.

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

Step 17.4.1.12.4

Rewrite the expression.

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

Step 17.4.1.13

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

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

Step 17.4.1.14

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

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Step 17.4.1.14.1

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

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

Step 17.4.1.14.2

Combine the numerators over the common denominator.

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

Step 17.4.1.14.3

Add $1$ and $3$ .

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

Step 17.4.1.14.4

Divide $4$ by $2$ .

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

Step 17.4.1.15

Move the negative in front of the fraction.

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

Step 17.4.2

Add $- \frac{1}{x}$ and $\frac{1}{x}$ .

$1 + 0 - \frac{1}{x^{2}}$

Step 17.4.3

Add $1$ and $0$ .

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