Calculus Examples

$f (\frac{\sqrt{6}}{3}) = \frac{1}{2} x^{4} - 2 x^{2}$

Step 1

Replace the variable $x$ with $\frac{\sqrt{6}}{3}$ in the expression.

$f (\frac{\sqrt{6}}{3}) = \frac{1}{2} \cdot {(\frac{\sqrt{6}}{3})}^{4} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $\frac{\sqrt{6}}{3}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{1}{2} \cdot \frac{{\sqrt{6}}^{4}}{3^{4}} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.2

Combine.

$f (\frac{\sqrt{6}}{3}) = \frac{1 {\sqrt{6}}^{4}}{2 \cdot 3^{4}} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.3

Multiply ${\sqrt{6}}^{4}$ by $1$ .

$f (\frac{\sqrt{6}}{3}) = \frac{{\sqrt{6}}^{4}}{2 \cdot 3^{4}} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.4

Raise $3$ to the power of $4$ .

$f (\frac{\sqrt{6}}{3}) = \frac{{\sqrt{6}}^{4}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 2.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{{(6^{\frac{1}{2}})}^{4}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.2

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

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{1}{2} \cdot 4}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{4}{2}}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.4

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

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Step 2.1.5.1.4.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{2 \cdot 2}{2}}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.4.2

Cancel the common factors.

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Step 2.1.5.1.4.2.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{2 \cdot 2}{2 (1)}}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{2 \cdot 2}{2 \cdot 1}}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{3}) = \frac{6^{\frac{2}{1}}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.1.4.2.4

Divide $2$ by $1$ .

$f (\frac{\sqrt{6}}{3}) = \frac{6^{2}}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.5.2

Raise $6$ to the power of $2$ .

$f (\frac{\sqrt{6}}{3}) = \frac{36}{2 \cdot 81} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.6

Multiply $2$ by $81$ .

$f (\frac{\sqrt{6}}{3}) = \frac{36}{162} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.7

Cancel the common factor of $36$ and $162$ .

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Step 2.1.7.1

Factor $18$ out of $36$ .

$f (\frac{\sqrt{6}}{3}) = \frac{18 (2)}{162} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.7.2

Cancel the common factors.

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Step 2.1.7.2.1

Factor $18$ out of $162$ .

$f (\frac{\sqrt{6}}{3}) = \frac{18 \cdot 2}{18 \cdot 9} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.7.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{3}) = \frac{18 \cdot 2}{18 \cdot 9} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.7.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 {(\frac{\sqrt{6}}{3})}^{2}$

Step 2.1.8

Apply the product rule to $\frac{\sqrt{6}}{3}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{{\sqrt{6}}^{2}}{3^{2}}$

Step 2.1.9

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{{(6^{\frac{1}{2}})}^{2}}{3^{2}}$

Step 2.1.9.2

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

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{6^{\frac{1}{2} \cdot 2}}{3^{2}}$

Step 2.1.9.3

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

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{6^{\frac{2}{2}}}{3^{2}}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{6^{\frac{2}{2}}}{3^{2}}$

Step 2.1.9.4.2

Rewrite the expression.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{6}{3^{2}}$

Step 2.1.9.5

Evaluate the exponent.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{6}{3^{2}}$

Step 2.1.10

Raise $3$ to the power of $2$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 (\frac{6}{9})$

Step 2.1.11

Cancel the common factor of $6$ and $9$ .

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Step 2.1.11.1

Factor $3$ out of $6$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{3 (2)}{9}$

Step 2.1.11.2

Cancel the common factors.

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Step 2.1.11.2.1

Factor $3$ out of $9$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{3 \cdot 2}{3 \cdot 3}$

Step 2.1.11.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 \frac{3 \cdot 2}{3 \cdot 3}$

Step 2.1.11.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - 2 (\frac{2}{3})$

Step 2.1.12

Multiply $- 2 (\frac{2}{3})$ .

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Step 2.1.12.1

Combine $- 2$ and $\frac{2}{3}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} + \frac{- 2 \cdot 2}{3}$

Step 2.1.12.2

Multiply $- 2$ by $2$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} + \frac{- 4}{3}$

Step 2.1.13

Move the negative in front of the fraction.

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - \frac{4}{3}$

Step 2.2

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - \frac{4}{3} \cdot \frac{3}{3}$

Step 2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4}{3}$ by $\frac{3}{3}$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - \frac{4 \cdot 3}{3 \cdot 3}$

Step 2.3.2

Multiply $3$ by $3$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2}{9} - \frac{4 \cdot 3}{9}$

Step 2.4

Combine the numerators over the common denominator.

$f (\frac{\sqrt{6}}{3}) = \frac{2 - 4 \cdot 3}{9}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 4$ by $3$ .

$f (\frac{\sqrt{6}}{3}) = \frac{2 - 12}{9}$

Step 2.5.2

Subtract $12$ from $2$ .

$f (\frac{\sqrt{6}}{3}) = \frac{- 10}{9}$

Step 2.6

Move the negative in front of the fraction.

$f (\frac{\sqrt{6}}{3}) = - \frac{10}{9}$

Step 2.7

The final answer is $- \frac{10}{9}$ .

$- \frac{10}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{10}{9}$

Decimal Form:

$- 1.\overline{1}$

Mixed Number Form:

$- 1 \frac{1}{9}$

Step 4