Calculus Examples

$(x^{\frac{1}{6}} + 1) (\frac{1}{6} x^{- \frac{1}{6}}) - (x^{\frac{1}{6}} - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1

Simplify each term.

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Step 1.1

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

$(x^{\frac{1}{6}} + 1) (\frac{1}{6} \cdot \frac{1}{x^{\frac{1}{6}}}) - (x^{\frac{1}{6}} - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.2

Combine.

$(x^{\frac{1}{6}} + 1) \frac{1 \cdot 1}{6 x^{\frac{1}{6}}} - (x^{\frac{1}{6}} - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.3

Multiply $1$ by $1$ .

$(x^{\frac{1}{6}} + 1) \frac{1}{6 x^{\frac{1}{6}}} - (x^{\frac{1}{6}} - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.4

Multiply $x^{\frac{1}{6}} + 1$ by $\frac{1}{6 x^{\frac{1}{6}}}$ .

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} - (x^{\frac{1}{6}} - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.5

Apply the distributive property.

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + (- x^{\frac{1}{6}} - - 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.6

Multiply $- 1$ by $- 1$ .

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + (- x^{\frac{1}{6}} + 1) (\frac{1}{6} x^{- \frac{1}{6}})$

Step 1.7

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

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + (- x^{\frac{1}{6}} + 1) (\frac{1}{6} \cdot \frac{1}{x^{\frac{1}{6}}})$

Step 1.8

Combine.

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + (- x^{\frac{1}{6}} + 1) \frac{1 \cdot 1}{6 x^{\frac{1}{6}}}$

Step 1.9

Multiply $1$ by $1$ .

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + (- x^{\frac{1}{6}} + 1) \frac{1}{6 x^{\frac{1}{6}}}$

Step 1.10

Multiply $- x^{\frac{1}{6}} + 1$ by $\frac{1}{6 x^{\frac{1}{6}}}$ .

$\frac{x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}} + \frac{- x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{6}} + 1 - x^{\frac{1}{6}} + 1}{6 x^{\frac{1}{6}}}$

Step 2.2

Combine the opposite terms in $x^{\frac{1}{6}} + 1 - x^{\frac{1}{6}} + 1$ .

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Step 2.2.1

Subtract $x^{\frac{1}{6}}$ from $x^{\frac{1}{6}}$ .

$\frac{0 + 1 + 1}{6 x^{\frac{1}{6}}}$

Step 2.2.2

Add $0$ and $1$ .

$\frac{1 + 1}{6 x^{\frac{1}{6}}}$

Step 2.3

Add $1$ and $1$ .

$\frac{2}{6 x^{\frac{1}{6}}}$

Step 2.4

Factor $2$ out of $2$ .

$\frac{2 (1)}{6 x^{\frac{1}{6}}}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $2$ out of $6 x^{\frac{1}{6}}$ .

$\frac{2 (1)}{2 (3 x^{\frac{1}{6}})}$

Step 3.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 (3 x^{\frac{1}{6}})}$

Step 3.3

Rewrite the expression.

$\frac{1}{3 x^{\frac{1}{6}}}$