Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 1.3

Multiply $1$ by $1$ .

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

Step 1.4

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

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- 1$ by $1$ .

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

Step 1.7

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

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

Step 1.8

Combine.

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

Step 1.9

Multiply $1$ by $1$ .

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

Step 1.10

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

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

Step 1.11

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

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Step 1.11.1

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

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

Step 1.11.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.11.3

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

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

Step 1.12

Move the negative in front of the fraction.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify by adding terms.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Add $0$ and $2$ .

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

Step 4.2

Subtract $1$ from $2$ .

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