Calculus Examples

$\frac{\sqrt{t}}{4 t - 3}$

Step 1

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

$\frac{d}{d t} [\frac{t^{\frac{1}{2}}}{4 t - 3}]$

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 $g (t) = 4 t - 3$ .

$\frac{(4 t - 3) \frac{d}{d t} [t^{\frac{1}{2}}] - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 3

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

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{1}{2} - 1}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 4

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

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 5

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

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 6

Combine the numerators over the common denominator.

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{1 - 2}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 7.2

Subtract $2$ from $1$ .

$\frac{(4 t - 3) (\frac{1}{2} t^{\frac{- 1}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$\frac{(4 t - 3) (\frac{1}{2} t^{- \frac{1}{2}}) - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 8.2

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

$\frac{(4 t - 3) \frac{t^{- \frac{1}{2}}}{2} - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 8.3

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

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} \frac{d}{d t} [4 t - 3]}{{(4 t - 3)}^{2}}$

Step 9

By the Sum Rule, the derivative of $4 t - 3$ with respect to $t$ is $\frac{d}{d t} [4 t] + \frac{d}{d t} [- 3]$ .

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (\frac{d}{d t} [4 t] + \frac{d}{d t} [- 3])}{{(4 t - 3)}^{2}}$

Step 10

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (4 \frac{d}{d t} [t] + \frac{d}{d t} [- 3])}{{(4 t - 3)}^{2}}$

Step 11

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

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (4 \cdot 1 + \frac{d}{d t} [- 3])}{{(4 t - 3)}^{2}}$

Step 12

Multiply $4$ by $1$ .

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (4 + \frac{d}{d t} [- 3])}{{(4 t - 3)}^{2}}$

Step 13

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

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} (4 + 0)}{{(4 t - 3)}^{2}}$

Step 14

Simplify the expression.

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Step 14.1

Add $4$ and $0$ .

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - t^{\frac{1}{2}} \cdot 4}{{(4 t - 3)}^{2}}$

Step 14.2

Multiply $4$ by $- 1$ .

$\frac{(4 t - 3) \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$\frac{4 t \frac{1}{2 t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2

Simplify the numerator.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $2$ .

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Step 15.2.1.1.1

Factor $2$ out of $4 t$ .

$\frac{2 (2 t) \frac{1}{2 t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.1.2

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

$\frac{2 (2 t) \frac{1}{2 (t^{\frac{1}{2}})} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.1.3

Cancel the common factor.

$\frac{2 (2 t) \frac{1}{2 t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.1.4

Rewrite the expression.

$\frac{2 t \frac{1}{t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.2

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

$\frac{t \frac{2}{t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.3

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

$\frac{\frac{t \cdot 2}{t^{\frac{1}{2}}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.4

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

$\frac{t \cdot 2 t^{- \frac{1}{2}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.5

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

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Step 15.2.1.5.1

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

$\frac{t^{- \frac{1}{2}} t \cdot 2 - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.5.2

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

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Step 15.2.1.5.2.1

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

$\frac{t^{- \frac{1}{2}} t^{1} \cdot 2 - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.5.2.2

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

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

Step 15.2.1.5.3

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

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

Step 15.2.1.5.4

Combine the numerators over the common denominator.

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

Step 15.2.1.5.5

Add $- 1$ and $2$ .

$\frac{t^{\frac{1}{2}} \cdot 2 - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.6

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

$\frac{2 \cdot t^{\frac{1}{2}} - 3 \frac{1}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.1.7

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

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

Step 15.2.1.8

Move the negative in front of the fraction.

$\frac{2 t^{\frac{1}{2}} - \frac{3}{2 t^{\frac{1}{2}}} - 4 t^{\frac{1}{2}}}{{(4 t - 3)}^{2}}$

Step 15.2.2

Subtract $4 t^{\frac{1}{2}}$ from $2 t^{\frac{1}{2}}$ .

$\frac{- 2 t^{\frac{1}{2}} - \frac{3}{2 t^{\frac{1}{2}}}}{{(4 t - 3)}^{2}}$

Step 15.3

Simplify the numerator.

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Step 15.3.1

Factor $- 1$ out of $- 2 t^{\frac{1}{2}} - \frac{3}{2 t^{\frac{1}{2}}}$ .

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Step 15.3.1.1

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

$\frac{- (2 t^{\frac{1}{2}}) - \frac{3}{2 t^{\frac{1}{2}}}}{{(4 t - 3)}^{2}}$

Step 15.3.1.2

Factor $- 1$ out of $- \frac{3}{2 t^{\frac{1}{2}}}$ .

$\frac{- (2 t^{\frac{1}{2}}) - (\frac{3}{2 t^{\frac{1}{2}}})}{{(4 t - 3)}^{2}}$

Step 15.3.1.3

Factor $- 1$ out of $- (2 t^{\frac{1}{2}}) - (\frac{3}{2 t^{\frac{1}{2}}})$ .

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

Step 15.3.2

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

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

Step 15.3.3

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

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

Step 15.3.4

Combine the numerators over the common denominator.

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

Step 15.3.5

Simplify the numerator.

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Step 15.3.5.1

Rewrite using the commutative property of multiplication.

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

Step 15.3.5.2

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

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Step 15.3.5.2.1

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

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

Step 15.3.5.2.2

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

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

Step 15.3.5.2.3

Combine the numerators over the common denominator.

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

Step 15.3.5.2.4

Add $1$ and $1$ .

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

Step 15.3.5.2.5

Divide $2$ by $2$ .

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

Step 15.3.5.3

Simplify $2 \cdot 2 t^{1}$ .

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

Step 15.3.5.4

Multiply $2$ by $2$ .

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

Step 15.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.5

Multiply $\frac{1}{{(4 t - 3)}^{2}}$ by $\frac{4 t + 3}{2 t^{\frac{1}{2}}}$ .

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

Step 15.6

Move $2$ to the left of ${(4 t - 3)}^{2}$ .

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

Step 15.7

Reorder factors in $- \frac{4 t + 3}{2 {(4 t - 3)}^{2} t^{\frac{1}{2}}}$ .

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