Calculus Examples

$(s^{- \frac{1}{2}} + 2 s) (7 - s^{- 1})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d s} [f (s) g (s)]$ is $f (s) \frac{d}{d s} [g (s)] + g (s) \frac{d}{d s} [f (s)]$ where $f (s) = s^{- \frac{1}{2}} + 2 s$ and $g (s) = 7 - s^{- 1}$ .

$(s^{- \frac{1}{2}} + 2 s) \frac{d}{d s} [7 - s^{- 1}] + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 - s^{- 1}$ with respect to $s$ is $\frac{d}{d s} [7] + \frac{d}{d s} [- s^{- 1}]$ .

$(s^{- \frac{1}{2}} + 2 s) (\frac{d}{d s} [7] + \frac{d}{d s} [- s^{- 1}]) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 2.2

Since $7$ is constant with respect to $s$ , the derivative of $7$ with respect to $s$ is $0$ .

$(s^{- \frac{1}{2}} + 2 s) (0 + \frac{d}{d s} [- s^{- 1}]) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 3

Evaluate $\frac{d}{d s} [- s^{- 1}]$ .

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Step 3.1

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

$(s^{- \frac{1}{2}} + 2 s) (0 - \frac{d}{d s} [s^{- 1}]) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 3.2

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

$(s^{- \frac{1}{2}} + 2 s) (0 - - s^{- 2}) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 3.3

Multiply $- 1$ by $- 1$ .

$(s^{- \frac{1}{2}} + 2 s) (0 + 1 s^{- 2}) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 3.4

Multiply $s^{- 2}$ by $1$ .

$(s^{- \frac{1}{2}} + 2 s) (0 + s^{- 2}) + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 4

Simplify.

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Step 4.1

Add $0$ and $s^{- 2}$ .

$(s^{- \frac{1}{2}} + 2 s) s^{- 2} + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 4.2

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) \frac{d}{d s} [s^{- \frac{1}{2}} + 2 s]$

Step 5

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (\frac{d}{d s} [s^{- \frac{1}{2}}] + \frac{d}{d s} [2 s])$

Step 6

Evaluate $\frac{d}{d s} [s^{- \frac{1}{2}}]$ .

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Step 6.1

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{1}{2} - 1} + \frac{d}{d s} [2 s])$

Step 6.2

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d s} [2 s])$

Step 6.3

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d s} [2 s])$

Step 6.4

Combine the numerators over the common denominator.

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{\frac{- 1 - 1 \cdot 2}{2}} + \frac{d}{d s} [2 s])$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 1$ by $2$ .

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{\frac{- 1 - 2}{2}} + \frac{d}{d s} [2 s])$

Step 6.5.2

Subtract $2$ from $- 1$ .

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{\frac{- 3}{2}} + \frac{d}{d s} [2 s])$

Step 6.6

Move the negative in front of the fraction.

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{3}{2}} + \frac{d}{d s} [2 s])$

Step 7

Evaluate $\frac{d}{d s} [2 s]$ .

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Step 7.1

Since $2$ is constant with respect to $s$ , the derivative of $2 s$ with respect to $s$ is $2 \frac{d}{d s} [s]$ .

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{3}{2}} + 2 \frac{d}{d s} [s])$

Step 7.2

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{3}{2}} + 2 \cdot 1)$

Step 7.3

Multiply $2$ by $1$ .

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} s^{- \frac{3}{2}} + 2)$

Step 8

Simplify each term.

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Step 8.1

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2} \cdot \frac{1}{s^{\frac{3}{2}}} + 2)$

Step 8.2

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{s^{\frac{3}{2}} \cdot 2} + 2)$

Step 8.3

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

$(s^{- \frac{1}{2}} + 2 s) \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$s^{- \frac{1}{2}} \frac{1}{s^{2}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2

Combine terms.

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Step 9.2.1

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

$\frac{s^{- \frac{1}{2}}}{s^{2}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.2

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

$\frac{1}{s^{2} s^{\frac{1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3

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

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Step 9.2.3.1

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

$\frac{1}{s^{2 + \frac{1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3.2

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

$\frac{1}{s^{2 \cdot \frac{2}{2} + \frac{1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3.3

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

$\frac{1}{s^{\frac{2 \cdot 2}{2} + \frac{1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3.4

Combine the numerators over the common denominator.

$\frac{1}{s^{\frac{2 \cdot 2 + 1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3.5

Simplify the numerator.

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Step 9.2.3.5.1

Multiply $2$ by $2$ .

$\frac{1}{s^{\frac{4 + 1}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.3.5.2

Add $4$ and $1$ .

$\frac{1}{s^{\frac{5}{2}}} + 2 s \frac{1}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.4

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

$\frac{1}{s^{\frac{5}{2}}} + s \frac{2}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.5

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

$\frac{1}{s^{\frac{5}{2}}} + \frac{s \cdot 2}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.6

Move $2$ to the left of $s$ .

$\frac{1}{s^{\frac{5}{2}}} + \frac{2 \cdot s}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.7

Cancel the common factor of $s$ and $s^{2}$ .

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Step 9.2.7.1

Factor $s$ out of $2 s$ .

$\frac{1}{s^{\frac{5}{2}}} + \frac{s \cdot 2}{s^{2}} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.7.2

Cancel the common factors.

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Step 9.2.7.2.1

Factor $s$ out of $s^{2}$ .

$\frac{1}{s^{\frac{5}{2}}} + \frac{s \cdot 2}{s \cdot s} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.7.2.2

Cancel the common factor.

$\frac{1}{s^{\frac{5}{2}}} + \frac{s \cdot 2}{s \cdot s} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.7.2.3

Rewrite the expression.

$\frac{1}{s^{\frac{5}{2}}} + \frac{2}{s} + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$

Step 9.2.8

To write $(7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2)$ as a fraction with a common denominator, multiply by $\frac{s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$ .

$\frac{2}{s} + \frac{1}{s^{\frac{5}{2}}} + \frac{(7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$

Step 9.2.9

Combine the numerators over the common denominator.

$\frac{2}{s} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$

Step 9.2.10

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

$\frac{2}{s} \cdot \frac{s^{\frac{5}{2}}}{s^{\frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$

Step 9.2.11

To write $\frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$ as a fraction with a common denominator, multiply by $\frac{s}{s}$ .

$\frac{2}{s} \cdot \frac{s^{\frac{5}{2}}}{s^{\frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12

Write each expression with a common denominator of $s \cdot s^{\frac{5}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.2.12.1

Multiply $\frac{2}{s}$ by $\frac{s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$ .

$\frac{2 s^{\frac{5}{2}}}{s \cdot s^{\frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.2

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

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Step 9.2.12.2.1

Multiply $s$ by $s^{\frac{5}{2}}$ .

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Step 9.2.12.2.1.1

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

$\frac{2 s^{\frac{5}{2}}}{s^{1} s^{\frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.2.1.2

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

$\frac{2 s^{\frac{5}{2}}}{s^{1 + \frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.2.2

Write $1$ as a fraction with a common denominator.

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{2}{2} + \frac{5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.2.3

Combine the numerators over the common denominator.

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{2 + 5}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.2.4

Add $2$ and $5$ .

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}} \cdot \frac{s}{s}$

Step 9.2.12.3

Multiply $\frac{1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$ by $\frac{s}{s}$ .

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{5}{2}} s}$

Step 9.2.12.4

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

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Step 9.2.12.4.1

Multiply $s^{\frac{5}{2}}$ by $s$ .

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Step 9.2.12.4.1.1

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

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{5}{2}} s^{1}}$

Step 9.2.12.4.1.2

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

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{5}{2} + 1}}$

Step 9.2.12.4.2

Write $1$ as a fraction with a common denominator.

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{5}{2} + \frac{2}{2}}}$

Step 9.2.12.4.3

Combine the numerators over the common denominator.

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{5 + 2}{2}}}$

Step 9.2.12.4.4

Add $5$ and $2$ .

$\frac{2 s^{\frac{5}{2}}}{s^{\frac{7}{2}}} + \frac{(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{7}{2}}}$

Step 9.2.13

Combine the numerators over the common denominator.

$\frac{2 s^{\frac{5}{2}} + (1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{7}{2}}}$

Step 9.2.14

Factor $s$ out of $2 s^{\frac{5}{2}} + (1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s$ .

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Step 9.2.14.1

Factor $s$ out of $2 s^{\frac{5}{2}}$ .

$\frac{s (2 s^{\frac{3}{2}}) + (1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s}{s^{\frac{7}{2}}}$

Step 9.2.14.2

Factor $s$ out of $(1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}) s$ .

$\frac{s (2 s^{\frac{3}{2}}) + s (1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}})}{s^{\frac{7}{2}}}$

Step 9.2.14.3

Factor $s$ out of $s (2 s^{\frac{3}{2}}) + s (1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}})$ .

$\frac{s (2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}})}{s^{\frac{7}{2}}}$

Step 9.2.15

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

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7}{2}} s^{- 1}}$

Step 9.2.16

Multiply $s^{\frac{7}{2}}$ by $s^{- 1}$ by adding the exponents.

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Step 9.2.16.1

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

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7}{2} - 1}}$

Step 9.2.16.2

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

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7}{2} - 1 \cdot \frac{2}{2}}}$

Step 9.2.16.3

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

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}}$

Step 9.2.16.4

Combine the numerators over the common denominator.

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7 - 1 \cdot 2}{2}}}$

Step 9.2.16.5

Simplify the numerator.

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Step 9.2.16.5.1

Multiply $- 1$ by $2$ .

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{7 - 2}{2}}}$

Step 9.2.16.5.2

Subtract $2$ from $7$ .

$\frac{2 s^{\frac{3}{2}} + 1 + (7 - s^{- 1}) (- \frac{1}{2 s^{\frac{3}{2}}} + 2) s^{\frac{5}{2}}}{s^{\frac{5}{2}}}$

Step 9.3

Reorder terms.

$\frac{s^{\frac{5}{2}} (- \frac{1}{2 s^{\frac{3}{2}}} + 2) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Apply the distributive property.

$\frac{(s^{\frac{5}{2}} (- \frac{1}{2 s^{\frac{3}{2}}}) + s^{\frac{5}{2}} \cdot 2) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.2

Rewrite using the commutative property of multiplication.

$\frac{(- s^{\frac{5}{2}} \frac{1}{2 s^{\frac{3}{2}}} + s^{\frac{5}{2}} \cdot 2) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.3

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

$\frac{(- s^{\frac{5}{2}} \frac{1}{2 s^{\frac{3}{2}}} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4

Simplify each term.

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Step 9.4.4.1

Cancel the common factor of $s^{\frac{3}{2}}$ .

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Step 9.4.4.1.1

Factor $s^{\frac{3}{2}}$ out of $- s^{\frac{5}{2}}$ .

$\frac{(s^{\frac{3}{2}} (- s^{\frac{2}{2}}) \frac{1}{2 s^{\frac{3}{2}}} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.1.2

Factor $s^{\frac{3}{2}}$ out of $2 s^{\frac{3}{2}}$ .

$\frac{(s^{\frac{3}{2}} (- s^{\frac{2}{2}}) \frac{1}{s^{\frac{3}{2}} \cdot 2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.1.3

Cancel the common factor.

$\frac{(s^{\frac{3}{2}} (- s^{\frac{2}{2}}) \frac{1}{s^{\frac{3}{2}} \cdot 2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.1.4

Rewrite the expression.

$\frac{(- s^{\frac{2}{2}} \frac{1}{2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.2

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

$\frac{(- \frac{s^{\frac{2}{2}}}{2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.3

Cancel the common factor of $2$ .

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Step 9.4.4.3.1

Cancel the common factor.

$\frac{(- \frac{s^{\frac{2}{2}}}{2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.3.2

Rewrite the expression.

$\frac{(- \frac{s^{1}}{2} + 2 \cdot s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.4.4

Simplify.

$\frac{(- \frac{s}{2} + 2 s^{\frac{5}{2}}) (7 - s^{- 1}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.5

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

$\frac{(- \frac{s}{2} + 2 s^{\frac{5}{2}}) (7 - \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.6

Expand $(- \frac{s}{2} + 2 s^{\frac{5}{2}}) (7 - \frac{1}{s})$ using the FOIL Method.

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Step 9.4.6.1

Apply the distributive property.

$\frac{- \frac{s}{2} (7 - \frac{1}{s}) + 2 s^{\frac{5}{2}} (7 - \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.6.2

Apply the distributive property.

$\frac{- \frac{s}{2} \cdot 7 - \frac{s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} (7 - \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.6.3

Apply the distributive property.

$\frac{- \frac{s}{2} \cdot 7 - \frac{s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7

Simplify each term.

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Step 9.4.7.1

Multiply $- \frac{s}{2} \cdot 7$ .

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Step 9.4.7.1.1

Multiply $7$ by $- 1$ .

$\frac{- 7 \frac{s}{2} - \frac{s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.1.2

Combine $- 7$ and $\frac{s}{2}$ .

$\frac{\frac{- 7 s}{2} - \frac{s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.2

Move the negative in front of the fraction.

$\frac{- \frac{7 s}{2} - \frac{s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.3

Cancel the common factor of $s$ .

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Step 9.4.7.3.1

Move the leading negative in $- \frac{s}{2}$ into the numerator.

$\frac{- \frac{7 s}{2} + \frac{- s}{2} (- \frac{1}{s}) + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.3.2

Move the leading negative in $- \frac{1}{s}$ into the numerator.

$\frac{- \frac{7 s}{2} + \frac{- s}{2} \cdot \frac{- 1}{s} + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.3.3

Factor $s$ out of $- s$ .

$\frac{- \frac{7 s}{2} + \frac{s \cdot - 1}{2} \cdot \frac{- 1}{s} + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.3.4

Cancel the common factor.

$\frac{- \frac{7 s}{2} + \frac{s \cdot - 1}{2} \cdot \frac{- 1}{s} + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.3.5

Rewrite the expression.

$\frac{- \frac{7 s}{2} + \frac{- 1}{2} \cdot - 1 + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.4

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

$\frac{- \frac{7 s}{2} + \frac{- - 1}{2} + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.5

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 2 s^{\frac{5}{2}} \cdot 7 + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.6

Multiply $7$ by $2$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + 2 s^{\frac{5}{2}} (- \frac{1}{s}) + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.7

Multiply $2 s^{\frac{5}{2}} (- \frac{1}{s})$ .

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Step 9.4.7.7.1

Multiply $- 1$ by $2$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} - 2 s^{\frac{5}{2}} \frac{1}{s} + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.7.2

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{5}{2}} \frac{- 2}{s} + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.7.3

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + \frac{s^{\frac{5}{2}} \cdot - 2}{s} + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.8

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{5}{2}} \cdot - 2 s^{- 1} + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9

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

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Step 9.4.7.9.1

Move $s^{- 1}$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{- 1} s^{\frac{5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.2

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{- 1 + \frac{5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.3

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{- 1 \cdot \frac{2}{2} + \frac{5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.4

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

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{- 1 \cdot 2}{2} + \frac{5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.5

Combine the numerators over the common denominator.

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{- 1 \cdot 2 + 5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.6

Simplify the numerator.

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Step 9.4.7.9.6.1

Multiply $- 1$ by $2$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{- 2 + 5}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.9.6.2

Add $- 2$ and $5$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} + s^{\frac{3}{2}} \cdot - 2 + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.7.10

Move $- 2$ to the left of $s^{\frac{3}{2}}$ .

$\frac{- \frac{7 s}{2} + \frac{1}{2} + 14 s^{\frac{5}{2}} - 2 s^{\frac{3}{2}} + 2 s^{\frac{3}{2}} + 1}{s^{\frac{5}{2}}}$

Step 9.4.8

Write $1$ as a fraction with a common denominator.

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} - 2 s^{\frac{3}{2}} + 2 s^{\frac{3}{2}} + \frac{1}{2} + \frac{2}{2}}{s^{\frac{5}{2}}}$

Step 9.4.9

Combine the numerators over the common denominator.

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} - 2 s^{\frac{3}{2}} + 2 s^{\frac{3}{2}} + \frac{1 + 2}{2}}{s^{\frac{5}{2}}}$

Step 9.4.10

Add $1$ and $2$ .

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} - 2 s^{\frac{3}{2}} + 2 s^{\frac{3}{2}} + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.11

Add $- 2 s^{\frac{3}{2}}$ and $2 s^{\frac{3}{2}}$ .

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} + 0 + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.12

Add $- \frac{7 s}{2}$ and $0$ .

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.13

To write $14 s^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{7 s}{2} + 14 s^{\frac{5}{2}} \cdot \frac{2}{2} + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.14

Combine $14 s^{\frac{5}{2}}$ and $\frac{2}{2}$ .

$\frac{- \frac{7 s}{2} + \frac{14 s^{\frac{5}{2}} \cdot 2}{2} + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.15

Combine the numerators over the common denominator.

$\frac{\frac{- 7 s + 14 s^{\frac{5}{2}} \cdot 2}{2} + \frac{3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.16

Combine the numerators over the common denominator.

$\frac{\frac{- 7 s + 14 s^{\frac{5}{2}} \cdot 2 + 3}{2}}{s^{\frac{5}{2}}}$

Step 9.4.17

Multiply $2$ by $14$ .

$\frac{\frac{- 7 s + 28 s^{\frac{5}{2}} + 3}{2}}{s^{\frac{5}{2}}}$

Step 9.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 7 s + 28 s^{\frac{5}{2}} + 3}{2} \cdot \frac{1}{s^{\frac{5}{2}}}$

Step 9.6

Multiply $\frac{- 7 s + 28 s^{\frac{5}{2}} + 3}{2}$ by $\frac{1}{s^{\frac{5}{2}}}$ .

$\frac{- 7 s + 28 s^{\frac{5}{2}} + 3}{2 s^{\frac{5}{2}}}$

Step 9.7

Factor $- 1$ out of $- 7 s$ .

$\frac{- (7 s) + 28 s^{\frac{5}{2}} + 3}{2 s^{\frac{5}{2}}}$

Step 9.8

Factor $- 1$ out of $28 s^{\frac{5}{2}}$ .

$\frac{- (7 s) - (- 28 s^{\frac{5}{2}}) + 3}{2 s^{\frac{5}{2}}}$

Step 9.9

Factor $- 1$ out of $- (7 s) - (- 28 s^{\frac{5}{2}})$ .

$\frac{- (7 s - 28 s^{\frac{5}{2}}) + 3}{2 s^{\frac{5}{2}}}$

Step 9.10

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (7 s - 28 s^{\frac{5}{2}}) - 1 (- 3)}{2 s^{\frac{5}{2}}}$

Step 9.11

Factor $- 1$ out of $- (7 s - 28 s^{\frac{5}{2}}) - 1 (- 3)$ .

$\frac{- (7 s - 28 s^{\frac{5}{2}} - 3)}{2 s^{\frac{5}{2}}}$

Step 9.12

Rewrite $- (7 s - 28 s^{\frac{5}{2}} - 3)$ as $- 1 (7 s - 28 s^{\frac{5}{2}} - 3)$ .

$\frac{- 1 (7 s - 28 s^{\frac{5}{2}} - 3)}{2 s^{\frac{5}{2}}}$

Step 9.13

Move the negative in front of the fraction.

$- \frac{7 s - 28 s^{\frac{5}{2}} - 3}{2 s^{\frac{5}{2}}}$