Calculus Examples

$p (t) = \frac{\sqrt{t}}{t - 1}$

Step 1

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

$\frac{d}{d t} [\frac{t^{\frac{1}{2}}}{t - 1}]$

Step 2

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 $f (t) = t^{\frac{1}{2}}$ and $g (t) = t - 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

$\frac{(t - 1) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (1 + 0)}{{(t - 1)}^{2}}$

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $0$ .

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

Step 12.2

Multiply $- 1$ by $1$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Simplify the numerator.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

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

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

Step 13.2.1.2

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

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

Step 13.2.1.3

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

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Step 13.2.1.3.1

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

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Step 13.2.1.3.1.1

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

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

Step 13.2.1.3.1.2

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

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

Step 13.2.1.3.2

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

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

Step 13.2.1.3.3

Combine the numerators over the common denominator.

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

Step 13.2.1.3.4

Subtract $1$ from $2$ .

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

Step 13.2.1.4

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

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

Step 13.2.2

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

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

Step 13.2.3

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

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

Step 13.2.4

Combine the numerators over the common denominator.

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

Step 13.2.5

Simplify each term.

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Step 13.2.5.1

Simplify the numerator.

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Step 13.2.5.1.1

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

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Step 13.2.5.1.1.1

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

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

Step 13.2.5.1.1.2

Multiply by $1$ .

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

Step 13.2.5.1.1.3

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

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

Step 13.2.5.1.1.4

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} \cdot 1 + t^{\frac{1}{2}} (- 1 \cdot 2)$ .

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

Step 13.2.5.1.2

Multiply $- 1$ by $2$ .

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

Step 13.2.5.1.3

Subtract $2$ from $1$ .

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

Step 13.2.5.2

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

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

Step 13.2.5.3

Move the negative in front of the fraction.

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

Step 13.3

Simplify the numerator.

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Step 13.3.1

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

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Step 13.3.1.1

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

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

Step 13.3.1.2

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

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

Step 13.3.1.3

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

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

Step 13.3.2

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

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

Step 13.3.3

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

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

Step 13.3.4

Combine the numerators over the common denominator.

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

Step 13.3.5

Simplify the numerator.

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Step 13.3.5.1

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

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Step 13.3.5.1.1

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

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

Step 13.3.5.1.2

Combine the numerators over the common denominator.

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

Step 13.3.5.1.3

Add $1$ and $1$ .

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

Step 13.3.5.1.4

Divide $2$ by $2$ .

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

Step 13.3.5.2

Simplify $t^{1}$ .

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

Step 13.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 13.5

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

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

Step 13.6

Move $2$ to the left of ${(t - 1)}^{2}$ .

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

Step 13.7

Reorder factors in $- \frac{t + 1}{2 {(t - 1)}^{2} t^{\frac{1}{2}}}$ .

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