Calculus Examples

$\frac{\sqrt{t}}{2 t - 1}$

Step 1

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

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

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 $g (t) = 2 t - 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Move the negative in front of the fraction.

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

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

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

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Multiply $2$ by $1$ .

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

Step 13

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

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

Step 14

Simplify the expression.

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Add $2$ and $0$ .

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

Multiply $2$ by $- 1$ .

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

Step 15

Simplify.

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Apply the distributive property.

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

Simplify the numerator.

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Simplify each term.

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Cancel the common factor of $2$ .

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Factor $2$ out of $2 t$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

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

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

Subtract $2 t^{\frac{1}{2}}$ from $t^{\frac{1}{2}}$ .

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

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $2 t^{1}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

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