Calculus Examples

$\frac{6 t}{6 + \sqrt{t}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d t} [\frac{6 t}{6 + t^{\frac{1}{2}}}]$

Step 1.2

Since $6$ is constant with respect to $t$ , the derivative of $\frac{6 t}{6 + t^{\frac{1}{2}}}$ with respect to $t$ is $6 \frac{d}{d t} [\frac{t}{6 + t^{\frac{1}{2}}}]$ .

$6 \frac{d}{d t} [\frac{t}{6 + t^{\frac{1}{2}}}]$

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 $f (t) = t$ and $g (t) = 6 + t^{\frac{1}{2}}$ .

$6 \frac{(6 + t^{\frac{1}{2}}) \frac{d}{d t} [t] - t \frac{d}{d t} [6 + t^{\frac{1}{2}}]}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$6 \frac{(6 + t^{\frac{1}{2}}) \cdot 1 - t \frac{d}{d t} [6 + t^{\frac{1}{2}}]}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3.2

Multiply $6 + t^{\frac{1}{2}}$ by $1$ .

$6 \frac{6 + t^{\frac{1}{2}} - t \frac{d}{d t} [6 + t^{\frac{1}{2}}]}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $6 + t^{\frac{1}{2}}$ with respect to $t$ is $\frac{d}{d t} [6] + \frac{d}{d t} [t^{\frac{1}{2}}]$ .

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{d}{d t} [6] + \frac{d}{d t} [t^{\frac{1}{2}}])}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3.4

Since $6$ is constant with respect to $t$ , the derivative of $6$ with respect to $t$ is $0$ .

$6 \frac{6 + t^{\frac{1}{2}} - t (0 + \frac{d}{d t} [t^{\frac{1}{2}}])}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d t} [t^{\frac{1}{2}}]$ .

$6 \frac{6 + t^{\frac{1}{2}} - t \frac{d}{d t} [t^{\frac{1}{2}}]}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 3.6

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

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{1}{2} - 1})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 4

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

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 5

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

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 6

Combine the numerators over the common denominator.

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{1 - 2}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 7.2

Subtract $2$ from $1$ .

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{\frac{- 1}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 8

Move the negative in front of the fraction.

$6 \frac{6 + t^{\frac{1}{2}} - t (\frac{1}{2} t^{- \frac{1}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 9

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

$6 \frac{6 + t^{\frac{1}{2}} - t \frac{t^{- \frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 10

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

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{- \frac{1}{2}} t}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 11

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

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Step 11.1

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

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Step 11.1.1

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

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{- \frac{1}{2}} t^{1}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 11.1.2

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

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{- \frac{1}{2} + 1}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 11.2

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

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{- \frac{1}{2} + \frac{2}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 11.3

Combine the numerators over the common denominator.

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{\frac{- 1 + 2}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 11.4

Add $- 1$ and $2$ .

$6 \frac{6 + t^{\frac{1}{2}} - \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 12

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

$6 \frac{6 + t^{\frac{1}{2}} \cdot \frac{2}{2} - \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 13

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

$6 \frac{6 + \frac{t^{\frac{1}{2}} \cdot 2}{2} - \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 14

Combine the numerators over the common denominator.

$6 \frac{6 + \frac{t^{\frac{1}{2}} \cdot 2 - t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 15

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

$6 \frac{6 + \frac{2 \cdot t^{\frac{1}{2}} - t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 16

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

$6 \frac{6 + \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 17

Multiply $\frac{6 + \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}}$ by $\frac{2}{2}$ .

$6 (\frac{2}{2} \cdot \frac{6 + \frac{t^{\frac{1}{2}}}{2}}{{(6 + t^{\frac{1}{2}})}^{2}})$

Step 18

Combine.

$6 \frac{2 (6 + \frac{t^{\frac{1}{2}}}{2})}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 19

Apply the distributive property.

$6 \frac{2 \cdot 6 + 2 \frac{t^{\frac{1}{2}}}{2}}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 20

Cancel the common factor of $2$ .

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Step 20.1

Cancel the common factor.

$6 \frac{2 \cdot 6 + 2 \frac{t^{\frac{1}{2}}}{2}}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 20.2

Rewrite the expression.

$6 \frac{2 \cdot 6 + t^{\frac{1}{2}}}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 21

Multiply $2$ by $6$ .

$6 \frac{12 + t^{\frac{1}{2}}}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 22

Combine $6$ and $\frac{12 + t^{\frac{1}{2}}}{2 {(6 + t^{\frac{1}{2}})}^{2}}$ .

$\frac{6 (12 + t^{\frac{1}{2}})}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 23

Factor $2$ out of $6 (12 + t^{\frac{1}{2}})$ .

$\frac{2 (3 (12 + t^{\frac{1}{2}}))}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 24

Cancel the common factors.

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Step 24.1

Factor $2$ out of $2 {(6 + t^{\frac{1}{2}})}^{2}$ .

$\frac{2 (3 (12 + t^{\frac{1}{2}}))}{2 ({(6 + t^{\frac{1}{2}})}^{2})}$

Step 24.2

Cancel the common factor.

$\frac{2 (3 (12 + t^{\frac{1}{2}}))}{2 {(6 + t^{\frac{1}{2}})}^{2}}$

Step 24.3

Rewrite the expression.

$\frac{3 (12 + t^{\frac{1}{2}})}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 25

Simplify.

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Step 25.1

Apply the distributive property.

$\frac{3 \cdot 12 + 3 t^{\frac{1}{2}}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 25.2

Multiply $3$ by $12$ .

$\frac{36 + 3 t^{\frac{1}{2}}}{{(6 + t^{\frac{1}{2}})}^{2}}$

Step 25.3

Reorder terms.

$\frac{3 t^{\frac{1}{2}} + 36}{{(t^{\frac{1}{2}} + 6)}^{2}}$

Step 25.4

Factor $3$ out of $3 t^{\frac{1}{2}} + 36$ .

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Step 25.4.1

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

$\frac{3 (t^{\frac{1}{2}}) + 36}{{(t^{\frac{1}{2}} + 6)}^{2}}$

Step 25.4.2

Factor $3$ out of $36$ .

$\frac{3 t^{\frac{1}{2}} + 3 \cdot 12}{{(t^{\frac{1}{2}} + 6)}^{2}}$

Step 25.4.3

Factor $3$ out of $3 t^{\frac{1}{2}} + 3 \cdot 12$ .

$\frac{3 (t^{\frac{1}{2}} + 12)}{{(t^{\frac{1}{2}} + 6)}^{2}}$