Calculus Examples

$\frac{1 - 14 t}{{(7 t)}^{\frac{1}{2}}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = 1 - 14 t$ and $g (t) = {(7 t)}^{\frac{1}{2}}$ .

$\frac{{(7 t)}^{\frac{1}{2}} \frac{d}{d t} [1 - 14 t] - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - 14 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- 14 t]$ .

$\frac{{(7 t)}^{\frac{1}{2}} (\frac{d}{d t} [1] + \frac{d}{d t} [- 14 t]) - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 2.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{{(7 t)}^{\frac{1}{2}} (0 + \frac{d}{d t} [- 14 t]) - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 3

Evaluate $\frac{d}{d t} [- 14 t]$ .

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Step 3.1

Since $- 14$ is constant with respect to $t$ , the derivative of $- 14 t$ with respect to $t$ is $- 14 \frac{d}{d t} [t]$ .

$\frac{{(7 t)}^{\frac{1}{2}} (0 - 14 \frac{d}{d t} [t]) - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 3.2

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

$\frac{{(7 t)}^{\frac{1}{2}} (0 - 14 \cdot 1) - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 3.3

Multiply $- 14$ by $1$ .

$\frac{{(7 t)}^{\frac{1}{2}} (0 - 14) - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 4

Differentiate.

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Step 4.1

Subtract $14$ from $0$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{d}{d t} [{(7 t)}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Factor $7$ out of $7 t$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{d}{d t} [{(7 (t))}^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 4.2.2

Apply the product rule to $7 (t)$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{d}{d t} [7^{\frac{1}{2}} t^{\frac{1}{2}}]}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 4.3

Since $7^{\frac{1}{2}}$ is constant with respect to $t$ , the derivative of $7^{\frac{1}{2}} t^{\frac{1}{2}}$ with respect to $t$ is $7^{\frac{1}{2}} \frac{d}{d t} [t^{\frac{1}{2}}]$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} \frac{d}{d t} [t^{\frac{1}{2}}])}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 4.4

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

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{1}{2} - 1}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 5

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

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 6

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

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 7

Combine the numerators over the common denominator.

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{1 - 2}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 8.2

Subtract $2$ from $1$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{\frac{- 1}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 9

Move the negative in front of the fraction.

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} (\frac{1}{2} t^{- \frac{1}{2}}))}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 10

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

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) (7^{\frac{1}{2}} \frac{t^{- \frac{1}{2}}}{2})}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 11

Combine $7^{\frac{1}{2}}$ and $\frac{t^{- \frac{1}{2}}}{2}$ .

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{7^{\frac{1}{2}} t^{- \frac{1}{2}}}{2}}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 12

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

$\frac{{(7 t)}^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 13

Simplify.

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Step 13.1

Apply the product rule to $7 t$ .

$\frac{7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{({(7 t)}^{\frac{1}{2}})}^{2}}$

Step 13.2

Apply the product rule to $7 t$ .

$\frac{7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}} t^{\frac{1}{2}})}^{2}}$

Step 13.3

Apply the product rule to $7^{\frac{1}{2}} t^{\frac{1}{2}}$ .

$\frac{7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot - 14 - (1 - 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.4

Apply the distributive property.

$\frac{7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot - 14 + (- 1 \cdot 1 - (- 14 t)) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.5

Apply the distributive property.

$\frac{7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot - 14 - 1 \cdot 1 \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} - (- 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6

Simplify each term.

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Step 13.6.1

Move $- 14$ to the left of $7^{\frac{1}{2}} t^{\frac{1}{2}}$ .

$\frac{- 14 \cdot (7^{\frac{1}{2}} t^{\frac{1}{2}}) - 1 \cdot 1 \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} - (- 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.2

Multiply $- 1$ by $1$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - 1 \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} - (- 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.3

Rewrite $- 1 \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ as $- \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} - (- 14 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.4

Multiply $- 14$ by $- 1$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 14 t \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.5

Cancel the common factor of $2$ .

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Step 13.6.5.1

Factor $2$ out of $14 t$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 2 (7 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.5.2

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 2 (7 t) \frac{7^{\frac{1}{2}}}{2 (t^{\frac{1}{2}})}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.5.3

Cancel the common factor.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 2 (7 t) \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.5.4

Rewrite the expression.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + 7 t \frac{7^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.6

Combine $7$ and $\frac{7^{\frac{1}{2}}}{t^{\frac{1}{2}}}$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7 \cdot 7^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.7

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

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Step 13.6.7.1

Multiply $7$ by $7^{\frac{1}{2}}$ .

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Step 13.6.7.1.1

Raise $7$ to the power of $1$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7^{1} \cdot 7^{\frac{1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.7.1.2

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7^{1 + \frac{1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.7.2

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7^{\frac{2}{2} + \frac{1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.7.3

Combine the numerators over the common denominator.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7^{\frac{2 + 1}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.7.4

Add $2$ and $1$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \frac{7^{\frac{3}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.8

Combine $t$ and $\frac{7^{\frac{3}{2}}}{t^{\frac{1}{2}}}$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + \frac{t \cdot 7^{\frac{3}{2}}}{t^{\frac{1}{2}}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.9

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t \cdot 7^{\frac{3}{2}} t^{- \frac{1}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10

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

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Step 13.6.10.1

Move $t^{- \frac{1}{2}}$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{- \frac{1}{2}} t \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10.2

Multiply $t^{- \frac{1}{2}}$ by $t$ .

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Step 13.6.10.2.1

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{- \frac{1}{2}} t^{1} \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10.2.2

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{- \frac{1}{2} + 1} \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10.3

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{- \frac{1}{2} + \frac{2}{2}} \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10.4

Combine the numerators over the common denominator.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{- 1 + 2}{2}} \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.6.10.5

Add $- 1$ and $2$ .

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{1}{2}} \cdot 7^{\frac{3}{2}}}{{(7^{\frac{1}{2}})}^{2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.7

Combine terms.

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Step 13.7.1

Multiply the exponents in ${(7^{\frac{1}{2}})}^{2}$ .

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Step 13.7.1.1

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

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{1}{2}} \cdot 7^{\frac{3}{2}}}{7^{\frac{1}{2} \cdot 2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.7.1.2

Cancel the common factor of $2$ .

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Step 13.7.1.2.1

Cancel the common factor.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{1}{2}} \cdot 7^{\frac{3}{2}}}{7^{\frac{1}{2} \cdot 2} {(t^{\frac{1}{2}})}^{2}}$

Step 13.7.1.2.2

Rewrite the expression.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{1}{2}} \cdot 7^{\frac{3}{2}}}{7^{1} {(t^{\frac{1}{2}})}^{2}}$

Step 13.7.2

Evaluate the exponent.

$\frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} + t^{\frac{1}{2}} \cdot 7^{\frac{3}{2}}}{7 {(t^{\frac{1}{2}})}^{2}}$

Step 13.7.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 13.7.3.1

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

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

Step 13.7.3.2

Cancel the common factor of $2$ .

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Step 13.7.3.2.1

Cancel the common factor.

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

Step 13.7.3.2.2

Rewrite the expression.

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

Step 13.7.4

Simplify.

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

Step 13.8

Reorder terms.

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

Step 13.9

Simplify the numerator.

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Step 13.9.1

To write $- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ .

$\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} - 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} \cdot \frac{2 t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.2

Combine $- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}}$ and $\frac{2 t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ .

$\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} + \frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} (2 t^{\frac{1}{2}})}{2 t^{\frac{1}{2}}} - \frac{7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.3

Combine the numerators over the common denominator.

$\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} + \frac{- 14 \cdot 7^{\frac{1}{2}} t^{\frac{1}{2}} (2 t^{\frac{1}{2}}) - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.4

Simplify the numerator.

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Step 13.9.4.1

Rewrite using the commutative property of multiplication.

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

Step 13.9.4.2

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

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Step 13.9.4.2.1

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

$\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} + \frac{- 14 \cdot 7^{\frac{1}{2}} \cdot 2 (t^{\frac{1}{2}} t^{\frac{1}{2}}) - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.4.2.2

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

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

Step 13.9.4.2.3

Combine the numerators over the common denominator.

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

Step 13.9.4.2.4

Add $1$ and $1$ .

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

Step 13.9.4.2.5

Divide $2$ by $2$ .

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

Step 13.9.4.3

Simplify $- 14 \cdot 7^{\frac{1}{2}} \cdot 2 t^{1}$ .

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

Step 13.9.4.4

Multiply $2$ by $- 14$ .

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

Step 13.9.5

To write $7^{\frac{3}{2}} t^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ .

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

Step 13.9.6

Combine $7^{\frac{3}{2}} t^{\frac{1}{2}}$ and $\frac{2 t^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ .

$\frac{\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} (2 t^{\frac{1}{2}})}{2 t^{\frac{1}{2}}} + \frac{- 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.7

Combine the numerators over the common denominator.

$\frac{\frac{7^{\frac{3}{2}} t^{\frac{1}{2}} (2 t^{\frac{1}{2}}) - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8

Simplify the numerator.

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Step 13.9.8.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{7^{\frac{3}{2}} \cdot 2 t^{\frac{1}{2}} t^{\frac{1}{2}} - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8.2

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

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Step 13.9.8.2.1

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

$\frac{\frac{7^{\frac{3}{2}} \cdot 2 (t^{\frac{1}{2}} t^{\frac{1}{2}}) - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8.2.2

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

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

Step 13.9.8.2.3

Combine the numerators over the common denominator.

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

Step 13.9.8.2.4

Add $1$ and $1$ .

$\frac{\frac{7^{\frac{3}{2}} \cdot 2 t^{\frac{2}{2}} - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8.2.5

Divide $2$ by $2$ .

$\frac{\frac{7^{\frac{3}{2}} \cdot 2 t^{1} - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8.3

Simplify $7^{\frac{3}{2}} \cdot 2 t^{1}$ .

$\frac{\frac{7^{\frac{3}{2}} \cdot 2 t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.9.8.4

Move $2$ to the left of $7^{\frac{3}{2}}$ .

$\frac{\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}}{7 t}$

Step 13.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} \cdot \frac{1}{7 t}$

Step 13.11

Multiply $\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}} \cdot \frac{1}{7 t}$ .

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Step 13.11.1

Multiply $\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}}}$ by $\frac{1}{7 t}$ .

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{2 t^{\frac{1}{2}} (7 t)}$

Step 13.11.2

Multiply $7$ by $2$ .

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{\frac{1}{2}} t}$

Step 13.11.3

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

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 (t^{1} t^{\frac{1}{2}})}$

Step 13.11.4

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

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{1 + \frac{1}{2}}}$

Step 13.11.5

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

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{\frac{2}{2} + \frac{1}{2}}}$

Step 13.11.6

Combine the numerators over the common denominator.

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{\frac{2 + 1}{2}}}$

Step 13.11.7

Add $2$ and $1$ .

$\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{\frac{3}{2}}}$

Step 13.12

Reorder factors in $\frac{2 \cdot 7^{\frac{3}{2}} t - 28 \cdot 7^{\frac{1}{2}} t - 7^{\frac{1}{2}}}{14 t^{\frac{3}{2}}}$ .

$\frac{2 t \cdot 7^{\frac{3}{2}} - 28 t \cdot 7^{\frac{1}{2}} - 7^{\frac{1}{2}}}{14 t^{\frac{3}{2}}}$