Calculus Examples

$f (x) = (u - \sqrt{u}) (u + \sqrt{u})$

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 $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d u} [f (u) g (u)]$ is $f (u) \frac{d}{d u} [g (u)] + g (u) \frac{d}{d u} [f (u)]$ where $f (u) = u - u^{\frac{1}{2}}$ and $g (u) = u + u^{\frac{1}{2}}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Combine the numerators over the common denominator.

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

Step 16

Simplify the numerator.

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Step 16.1

Multiply $- 1$ by $2$ .

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

Step 16.2

Subtract $2$ from $1$ .

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

Step 17

Move the negative in front of the fraction.

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

Step 18

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

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

Step 19

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

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

Step 20

Simplify.

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Step 20.1

Reorder terms.

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

Step 20.2

Simplify each term.

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Step 20.2.1

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

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Step 20.2.1.1

Apply the distributive property.

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

Step 20.2.1.2

Apply the distributive property.

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

Step 20.2.1.3

Apply the distributive property.

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

Step 20.2.2

Simplify and combine like terms.

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Step 20.2.2.1

Simplify each term.

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Step 20.2.2.1.1

Multiply $u$ by $1$ .

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

Step 20.2.2.1.2

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

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

Step 20.2.2.1.3

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

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

Step 20.2.2.1.4

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

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

Step 20.2.2.1.5

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

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Step 20.2.2.1.5.1

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

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Step 20.2.2.1.5.1.1

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

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

Step 20.2.2.1.5.1.2

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

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

Step 20.2.2.1.5.2

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

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

Step 20.2.2.1.5.3

Combine the numerators over the common denominator.

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

Step 20.2.2.1.5.4

Subtract $1$ from $2$ .

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

Step 20.2.2.1.6

Rewrite using the commutative property of multiplication.

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

Step 20.2.2.1.7

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

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Step 20.2.2.1.7.1

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

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

Step 20.2.2.1.7.2

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

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

Step 20.2.2.1.7.3

Cancel the common factor.

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

Step 20.2.2.1.7.4

Rewrite the expression.

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

Step 20.2.2.1.8

Move the negative in front of the fraction.

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

Step 20.2.2.2

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

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

Step 20.2.2.3

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

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

Step 20.2.2.4

Combine the numerators over the common denominator.

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

Step 20.2.2.5

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

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

Step 20.2.2.6

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

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

Step 20.2.2.7

Combine the numerators over the common denominator.

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

Step 20.2.2.8

Combine the numerators over the common denominator.

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

Step 20.2.3

Simplify the numerator.

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Step 20.2.3.1

Move $2$ to the left of $u$ .

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

Step 20.2.3.2

Multiply $2$ by $- 1$ .

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

Step 20.2.3.3

Add $- 2 u^{\frac{1}{2}}$ and $u^{\frac{1}{2}}$ .

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

Step 20.2.3.4

Rewrite $2 u - u^{\frac{1}{2}} - 1$ in a factored form.

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Step 20.2.3.4.1

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

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

Step 20.2.3.4.2

Let $u = u^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $u^{\frac{1}{2}}$ .

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

Step 20.2.3.4.3

Factor by grouping.

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Step 20.2.3.4.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 20.2.3.4.3.1.1

Factor $- 1$ out of $- u$ .

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

Step 20.2.3.4.3.1.2

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 20.2.3.4.3.1.3

Apply the distributive property.

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

Step 20.2.3.4.3.1.4

Multiply $u$ by $1$ .

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

Step 20.2.3.4.3.2

Factor out the greatest common factor from each group.

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Step 20.2.3.4.3.2.1

Group the first two terms and the last two terms.

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

Step 20.2.3.4.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 20.2.3.4.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

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

Step 20.2.3.4.4

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

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

Step 20.2.4

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

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Step 20.2.4.1

Apply the distributive property.

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

Step 20.2.4.2

Apply the distributive property.

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

Step 20.2.4.3

Apply the distributive property.

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

Step 20.2.5

Simplify and combine like terms.

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Step 20.2.5.1

Simplify each term.

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Step 20.2.5.1.1

Multiply $u$ by $1$ .

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

Step 20.2.5.1.2

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

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

Step 20.2.5.1.3

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

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

Step 20.2.5.1.4

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

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

Step 20.2.5.1.5

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

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Step 20.2.5.1.5.1

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

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Step 20.2.5.1.5.1.1

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

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

Step 20.2.5.1.5.1.2

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

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

Step 20.2.5.1.5.2

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

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

Step 20.2.5.1.5.3

Combine the numerators over the common denominator.

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

Step 20.2.5.1.5.4

Subtract $1$ from $2$ .

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

Step 20.2.5.1.6

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

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Step 20.2.5.1.6.1

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

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

Step 20.2.5.1.6.2

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

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

Step 20.2.5.1.6.3

Cancel the common factor.

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

Step 20.2.5.1.6.4

Rewrite the expression.

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

Step 20.2.5.1.7

Move the negative in front of the fraction.

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

Step 20.2.5.2

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

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

Step 20.2.5.3

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

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

Step 20.2.5.4

Combine the numerators over the common denominator.

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

Step 20.2.6

Combine the numerators over the common denominator.

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

Step 20.2.7

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

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

Step 20.2.8

Subtract $u^{\frac{1}{2}}$ from $2 u^{\frac{1}{2}}$ .

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

Step 20.2.9

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

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

Step 20.2.10

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

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

Step 20.2.11

Combine the numerators over the common denominator.

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

Step 20.2.12

Simplify the numerator.

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Step 20.2.12.1

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

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

Step 20.2.12.2

Let $u = u^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $u^{\frac{1}{2}}$ .

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

Step 20.2.12.3

Factor by grouping.

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Step 20.2.12.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 20.2.12.3.1.1

Multiply by $1$ .

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

Step 20.2.12.3.1.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 20.2.12.3.1.3

Apply the distributive property.

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

Step 20.2.12.3.2

Factor out the greatest common factor from each group.

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Step 20.2.12.3.2.1

Group the first two terms and the last two terms.

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

Step 20.2.12.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 20.2.12.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

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

Step 20.2.12.4

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

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

Step 20.3

Combine the numerators over the common denominator.

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

Step 20.4

Simplify each term.

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Step 20.4.1

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

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Step 20.4.1.1

Apply the distributive property.

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

Step 20.4.1.2

Apply the distributive property.

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

Step 20.4.1.3

Apply the distributive property.

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

Step 20.4.2

Simplify and combine like terms.

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Step 20.4.2.1

Simplify each term.

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Step 20.4.2.1.1

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

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Step 20.4.2.1.1.1

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

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

Step 20.4.2.1.1.2

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

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

Step 20.4.2.1.1.3

Combine the numerators over the common denominator.

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

Step 20.4.2.1.1.4

Add $1$ and $1$ .

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

Step 20.4.2.1.1.5

Divide $2$ by $2$ .

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

Step 20.4.2.1.2

Simplify $2 u^{1}$ .

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

Step 20.4.2.1.3

Multiply $- 1$ by $2$ .

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

Step 20.4.2.1.4

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

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

Step 20.4.2.1.5

Multiply $- 1$ by $1$ .

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

Step 20.4.2.2

Add $- 2 u^{\frac{1}{2}}$ and $u^{\frac{1}{2}}$ .

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

Step 20.4.3

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

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Step 20.4.3.1

Apply the distributive property.

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

Step 20.4.3.2

Apply the distributive property.

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

Step 20.4.3.3

Apply the distributive property.

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u^{\frac{1}{2}} u^{\frac{1}{2}} + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4

Simplify and combine like terms.

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Step 20.4.4.1

Simplify each term.

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Step 20.4.4.1.1

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

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Step 20.4.4.1.1.1

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

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

Step 20.4.4.1.1.2

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

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u^{\frac{1}{2} + \frac{1}{2}} + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.1.3

Combine the numerators over the common denominator.

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u^{\frac{1 + 1}{2}} + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.1.4

Add $1$ and $1$ .

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u^{\frac{2}{2}} + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.1.5

Divide $2$ by $2$ .

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u^{1} + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.2

Simplify $2 u^{1}$ .

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u + 2 u^{\frac{1}{2}} \cdot 1 - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.3

Multiply $2$ by $1$ .

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u + 2 u^{\frac{1}{2}} - 1 u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.4

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

$\frac{2 u - u^{\frac{1}{2}} - 1 + 2 u + 2 u^{\frac{1}{2}} - u^{\frac{1}{2}} - 1 \cdot 1}{2}$

Step 20.4.4.1.5

Multiply $- 1$ by $1$ .

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

Step 20.4.4.2

Subtract $u^{\frac{1}{2}}$ from $2 u^{\frac{1}{2}}$ .

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

Step 20.5

Combine the opposite terms in $2 u - u^{\frac{1}{2}} - 1 + 2 u + u^{\frac{1}{2}} - 1$ .

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Step 20.5.1

Add $- u^{\frac{1}{2}}$ and $u^{\frac{1}{2}}$ .

$\frac{2 u - 1 + 2 u + 0 - 1}{2}$

Step 20.5.2

Add $2 u - 1 + 2 u$ and $0$ .

$\frac{2 u - 1 + 2 u - 1}{2}$

Step 20.6

Add $2 u$ and $2 u$ .

$\frac{4 u - 1 - 1}{2}$

Step 20.7

Subtract $1$ from $- 1$ .

$\frac{4 u - 2}{2}$

Step 20.8

Cancel the common factor of $4 u - 2$ and $2$ .

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Step 20.8.1

Factor $2$ out of $4 u$ .

$\frac{2 (2 u) - 2}{2}$

Step 20.8.2

Factor $2$ out of $- 2$ .

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

Step 20.8.3

Factor $2$ out of $2 (2 u) + 2 (- 1)$ .

$\frac{2 (2 u - 1)}{2}$

Step 20.8.4

Cancel the common factors.

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Step 20.8.4.1

Factor $2$ out of $2$ .

$\frac{2 (2 u - 1)}{2 (1)}$

Step 20.8.4.2

Cancel the common factor.

$\frac{2 (2 u - 1)}{2 \cdot 1}$

Step 20.8.4.3

Rewrite the expression.

$\frac{2 u - 1}{1}$

Step 20.8.4.4

Divide $2 u - 1$ by $1$ .

$2 u - 1$