Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 1.3

Multiply $1$ by $1$ .

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

Step 1.4

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

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite $1$ as $1^{3}$ .

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

Step 1.5.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $- 1$ by $1$ .

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

Step 1.5.3.2

One to any power is one.

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

Step 1.6

Rewrite using the commutative property of multiplication.

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

Step 1.7

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

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Step 1.7.1

Move $x^{2}$ .

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.7.4

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

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

Step 1.7.5

Combine the numerators over the common denominator.

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

Step 1.7.6

Simplify the numerator.

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Step 1.7.6.1

Multiply $2$ by $2$ .

$\frac{(x + 1) (x^{2} - x + 1)}{2 x^{\frac{1}{2}}} - 1 \cdot 3 x^{\frac{4 + 1}{2}}$

Step 1.7.6.2

Add $4$ and $1$ .

$\frac{(x + 1) (x^{2} - x + 1)}{2 x^{\frac{1}{2}}} - 1 \cdot 3 x^{\frac{5}{2}}$

Step 1.8

Multiply $- 1$ by $3$ .

$\frac{(x + 1) (x^{2} - x + 1)}{2 x^{\frac{1}{2}}} - 3 x^{\frac{5}{2}}$

Step 2

To write $- 3 x^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{(x + 1) (x^{2} - x + 1)}{2 x^{\frac{1}{2}}} - 3 x^{\frac{5}{2}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 3 x^{\frac{5}{2}}$ and $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{(x + 1) (x^{2} - x + 1)}{2 x^{\frac{1}{2}}} + \frac{- 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(x + 1) (x^{2} - x + 1) - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x + 1) (x^{2} - x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x \cdot x^{2} + x (- x) + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply $x$ by $x^{2}$ .

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Step 4.2.1.1.1

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

$\frac{x^{1} x^{2} + x (- x) + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.1.1.2

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

$\frac{x^{1 + 2} + x (- x) + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.1.2

Add $1$ and $2$ .

$\frac{x^{3} + x (- x) + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.2

Rewrite using the commutative property of multiplication.

$\frac{x^{3} - x \cdot x + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.3.1

Move $x$ .

$\frac{x^{3} - (x \cdot x) + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.3.2

Multiply $x$ by $x$ .

$\frac{x^{3} - x^{2} + x \cdot 1 + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.4

Multiply $x$ by $1$ .

$\frac{x^{3} - x^{2} + x + 1 x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.5

Multiply $x^{2}$ by $1$ .

$\frac{x^{3} - x^{2} + x + x^{2} + 1 (- x) + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.6

Multiply $- x$ by $1$ .

$\frac{x^{3} - x^{2} + x + x^{2} - x + 1 \cdot 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.2.7

Multiply $1$ by $1$ .

$\frac{x^{3} - x^{2} + x + x^{2} - x + 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.3

Combine the opposite terms in $x^{3} - x^{2} + x + x^{2} - x + 1$ .

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Step 4.3.1

Add $- x^{2}$ and $x^{2}$ .

$\frac{x^{3} + x + 0 - x + 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.3.2

Add $x^{3} + x$ and $0$ .

$\frac{x^{3} + x - x + 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.3.3

Subtract $x$ from $x$ .

$\frac{x^{3} + 0 + 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.3.4

Add $x^{3}$ and $0$ .

$\frac{x^{3} + 1 - 3 x^{\frac{5}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.4

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + 1 - 3 \cdot 2 x^{\frac{5}{2}} x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 4.5

Multiply $x^{\frac{5}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.5.1

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

$\frac{x^{3} + 1 - 3 \cdot 2 (x^{\frac{1}{2}} x^{\frac{5}{2}})}{2 x^{\frac{1}{2}}}$

Step 4.5.2

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

$\frac{x^{3} + 1 - 3 \cdot 2 x^{\frac{1}{2} + \frac{5}{2}}}{2 x^{\frac{1}{2}}}$

Step 4.5.3

Combine the numerators over the common denominator.

$\frac{x^{3} + 1 - 3 \cdot 2 x^{\frac{1 + 5}{2}}}{2 x^{\frac{1}{2}}}$

Step 4.5.4

Add $1$ and $5$ .

$\frac{x^{3} + 1 - 3 \cdot 2 x^{\frac{6}{2}}}{2 x^{\frac{1}{2}}}$

Step 4.5.5

Divide $6$ by $2$ .

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

Step 4.6

Multiply $- 3$ by $2$ .

$\frac{x^{3} + 1 - 6 x^{3}}{2 x^{\frac{1}{2}}}$

Step 4.7

Subtract $6 x^{3}$ from $x^{3}$ .

$\frac{- 5 x^{3} + 1}{2 x^{\frac{1}{2}}}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 5 x^{3}$ .

$\frac{- (5 x^{3}) + 1}{2 x^{\frac{1}{2}}}$

Step 5.2

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

$\frac{- (5 x^{3}) - 1 (- 1)}{2 x^{\frac{1}{2}}}$

Step 5.3

Factor $- 1$ out of $- (5 x^{3}) - 1 (- 1)$ .

$\frac{- (5 x^{3} - 1)}{2 x^{\frac{1}{2}}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (5 x^{3} - 1)$ as $- 1 (5 x^{3} - 1)$ .

$\frac{- 1 (5 x^{3} - 1)}{2 x^{\frac{1}{2}}}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{5 x^{3} - 1}{2 x^{\frac{1}{2}}}$