Calculus Examples

$f (x) = 5 \sqrt{x} (x^{3} - 5 \sqrt{x} + 3)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [5 x^{\frac{1}{2}} (x^{3} - 5 \sqrt{x} + 3)]$

Step 1.2

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

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

Step 1.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{1}{2}} (x^{3} - 5 x^{\frac{1}{2}} + 3)$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{1}{2}} (x^{3} - 5 x^{\frac{1}{2}} + 3)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{3} - 5 x^{\frac{1}{2}} + 3$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{3} - 5 x^{\frac{1}{2}} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}] + \frac{d}{d x} [3]$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{\frac{1}{2}}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.4

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Simplify the expression.

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Step 8.4.1

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

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

Step 8.4.2

Move the negative in front of the fraction.

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

Step 9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 10

Add $3 x^{2} - \frac{5}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Combine the numerators over the common denominator.

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

Step 15

Simplify the numerator.

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Step 15.1

Multiply $- 1$ by $2$ .

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

Step 15.2

Subtract $2$ from $1$ .

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

Step 16

Move the negative in front of the fraction.

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

Step 17

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

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

Step 18

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

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

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

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

Step 19.2

Apply the distributive property.

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

Step 19.3

Apply the distributive property.

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

Step 19.4

Combine terms.

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Step 19.4.1

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

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Step 19.4.1.1

Move $x^{2}$ .

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

Step 19.4.1.2

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

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

Step 19.4.1.3

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

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

Step 19.4.1.4

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

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

Step 19.4.1.5

Combine the numerators over the common denominator.

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

Step 19.4.1.6

Simplify the numerator.

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Step 19.4.1.6.1

Multiply $2$ by $2$ .

$5 (x^{\frac{4 + 1}{2}} \cdot 3) + 5 (x^{\frac{1}{2}} (- \frac{5}{2 x^{\frac{1}{2}}})) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.1.6.2

Add $4$ and $1$ .

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

Step 19.4.2

Move $3$ to the left of $x^{\frac{5}{2}}$ .

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

Step 19.4.3

Multiply $3$ by $5$ .

$15 x^{\frac{5}{2}} + 5 (x^{\frac{1}{2}} (- \frac{5}{2 x^{\frac{1}{2}}})) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.4

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

$15 x^{\frac{5}{2}} + 5 (- \frac{x^{\frac{1}{2}} \cdot 5}{2 x^{\frac{1}{2}}}) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.5

Move $5$ to the left of $x^{\frac{1}{2}}$ .

$15 x^{\frac{5}{2}} + 5 (- \frac{5 \cdot x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.6

Cancel the common factor.

$15 x^{\frac{5}{2}} + 5 (- \frac{5 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.7

Rewrite the expression.

$15 x^{\frac{5}{2}} + 5 (- \frac{5}{2}) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.8

Multiply $- 1$ by $5$ .

$15 x^{\frac{5}{2}} - 5 (\frac{5}{2}) + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.9

Combine $- 5$ and $\frac{5}{2}$ .

$15 x^{\frac{5}{2}} + \frac{- 5 \cdot 5}{2} + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.10

Multiply $- 5$ by $5$ .

$15 x^{\frac{5}{2}} + \frac{- 25}{2} + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.11

Move the negative in front of the fraction.

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 (x^{3} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.12

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{3}}{2 x^{\frac{1}{2}}} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.13

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{3} x^{- \frac{1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14

Multiply $x^{3}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 19.4.14.1

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{3 - \frac{1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{3 \cdot \frac{2}{2} - \frac{1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14.3

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{\frac{3 \cdot 2}{2} - \frac{1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14.4

Combine the numerators over the common denominator.

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{\frac{3 \cdot 2 - 1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14.5

Simplify the numerator.

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Step 19.4.14.5.1

Multiply $3$ by $2$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{\frac{6 - 1}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.14.5.2

Subtract $1$ from $6$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + 5 \frac{x^{\frac{5}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.15

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 (- 5 x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.16

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 (x^{\frac{1}{2}} \frac{- 5}{2 x^{\frac{1}{2}}}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.17

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 \frac{x^{\frac{1}{2}} \cdot - 5}{2 x^{\frac{1}{2}}} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.18

Move $- 5$ to the left of $x^{\frac{1}{2}}$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 \frac{- 5 \cdot x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.19

Cancel the common factor.

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 \frac{- 5 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.20

Rewrite the expression.

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 (\frac{- 5}{2}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.21

Move the negative in front of the fraction.

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + 5 (- \frac{5}{2}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.22

Multiply $- 1$ by $5$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} - 5 (\frac{5}{2}) + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.23

Combine $- 5$ and $\frac{5}{2}$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + \frac{- 5 \cdot 5}{2} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.24

Multiply $- 5$ by $5$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} + \frac{- 25}{2} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.25

Move the negative in front of the fraction.

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} + 5 (3 \frac{1}{2 x^{\frac{1}{2}}})$

Step 19.4.26

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} + 5 \frac{3}{2 x^{\frac{1}{2}}}$

Step 19.4.27

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

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} + \frac{5 \cdot 3}{2 x^{\frac{1}{2}}}$

Step 19.4.28

Multiply $5$ by $3$ .

$15 x^{\frac{5}{2}} - \frac{25}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.29

To write $15 x^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$15 x^{\frac{5}{2}} \cdot \frac{2}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.30

Combine $15 x^{\frac{5}{2}}$ and $\frac{2}{2}$ .

$\frac{15 x^{\frac{5}{2}} \cdot 2}{2} + \frac{5 x^{\frac{5}{2}}}{2} - \frac{25}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.31

Combine the numerators over the common denominator.

$\frac{15 x^{\frac{5}{2}} \cdot 2 + 5 x^{\frac{5}{2}}}{2} - \frac{25}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.32

Multiply $2$ by $15$ .

$\frac{30 x^{\frac{5}{2}} + 5 x^{\frac{5}{2}}}{2} - \frac{25}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.33

Add $30 x^{\frac{5}{2}}$ and $5 x^{\frac{5}{2}}$ .

$\frac{35 x^{\frac{5}{2}}}{2} - \frac{25}{2} - \frac{25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.34

Subtract $\frac{25}{2}$ from $- \frac{25}{2}$ .

$\frac{35 x^{\frac{5}{2}}}{2} - 2 (\frac{25}{2}) + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.35

Combine $- 2$ and $\frac{25}{2}$ .

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{- 2 \cdot 25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.36

Multiply $- 2$ by $25$ .

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{- 50}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.37

Cancel the common factor of $- 50$ and $2$ .

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Step 19.4.37.1

Factor $2$ out of $- 50$ .

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{2 \cdot - 25}{2} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.37.2

Cancel the common factors.

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Step 19.4.37.2.1

Factor $2$ out of $2$ .

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{2 \cdot - 25}{2 (1)} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.37.2.2

Cancel the common factor.

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{2 \cdot - 25}{2 \cdot 1} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.37.2.3

Rewrite the expression.

$\frac{35 x^{\frac{5}{2}}}{2} + \frac{- 25}{1} + \frac{15}{2 x^{\frac{1}{2}}}$

Step 19.4.37.2.4

Divide $- 25$ by $1$ .

$\frac{35 x^{\frac{5}{2}}}{2} - 25 + \frac{15}{2 x^{\frac{1}{2}}}$