Calculus Examples

$\frac{4 t}{\sqrt[3]{t^{2} + 3}}$

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 $\sqrt[3]{t^{2} + 3}$ as ${(t^{2} + 3)}^{\frac{1}{3}}$ .

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

Step 1.2

Since $4$ is constant with respect to $t$ , the derivative of $\frac{4 t}{{(t^{2} + 3)}^{\frac{1}{3}}}$ with respect to $t$ is $4 \frac{d}{d t} [\frac{t}{{(t^{2} + 3)}^{\frac{1}{3}}}]$ .

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

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 $f (t) = t$ and $g (t) = {(t^{2} + 3)}^{\frac{1}{3}}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(t^{2} + 3)}^{\frac{1}{3}})}^{2}$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{3}}$ and $g (t) = t^{2} + 3$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $t^{2} + 3$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{3}$ .

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

Step 4.3

Replace all occurrences of $u$ with $t^{2} + 3$ .

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

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

Step 8.2

Subtract $3$ from $1$ .

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

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

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

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

Step 9.3

Move ${(t^{2} + 3)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.4

Combine $\frac{1}{3 {(t^{2} + 3)}^{\frac{2}{3}}}$ and $t$ .

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

Step 10

By the Sum Rule, the derivative of $t^{2} + 3$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [3]$ .

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

Step 11

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

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

Step 12

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

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

Step 13

Combine fractions.

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Step 13.1

Add $2 t$ and $0$ .

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

Step 13.2

Multiply $2$ by $- 1$ .

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

Step 13.3

Combine $- 2$ and $\frac{t}{3 {(t^{2} + 3)}^{\frac{2}{3}}}$ .

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

Step 13.4

Combine $\frac{- 2 t}{3 {(t^{2} + 3)}^{\frac{2}{3}}}$ and $t$ .

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Add $1$ and $1$ .

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

Step 18

Move the negative in front of the fraction.

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

Step 19

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

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

Step 20

Combine ${(t^{2} + 3)}^{\frac{1}{3}}$ and $\frac{3 {(t^{2} + 3)}^{\frac{2}{3}}}{3 {(t^{2} + 3)}^{\frac{2}{3}}}$ .

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

Step 21

Combine the numerators over the common denominator.

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

Step 22

Multiply ${(t^{2} + 3)}^{\frac{1}{3}}$ by ${(t^{2} + 3)}^{\frac{2}{3}}$ by adding the exponents.

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Step 22.1

Move ${(t^{2} + 3)}^{\frac{2}{3}}$ .

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

Step 22.2

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

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

Step 22.3

Combine the numerators over the common denominator.

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

Step 22.4

Add $2$ and $1$ .

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

Step 22.5

Divide $3$ by $3$ .

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

Step 23

Simplify ${(t^{2} + 3)}^{1} \cdot 3$ .

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

Step 24

Move $3$ to the left of $t^{2} + 3$ .

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

Step 25

Rewrite $\frac{\frac{3 (t^{2} + 3) - 2 t^{2}}{3 {(t^{2} + 3)}^{\frac{2}{3}}}}{{(t^{2} + 3)}^{\frac{2}{3}}}$ as a product.

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

Step 26

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

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

Step 27

Multiply ${(t^{2} + 3)}^{\frac{2}{3}}$ by ${(t^{2} + 3)}^{\frac{2}{3}}$ by adding the exponents.

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Step 27.1

Move ${(t^{2} + 3)}^{\frac{2}{3}}$ .

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

Step 27.2

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

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

Step 27.3

Combine the numerators over the common denominator.

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

Step 27.4

Add $2$ and $2$ .

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

Step 28

Combine $4$ and $\frac{3 (t^{2} + 3) - 2 t^{2}}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$ .

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

Step 29

Simplify.

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Step 29.1

Apply the distributive property.

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

Step 29.2

Apply the distributive property.

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

Step 29.3

Simplify the numerator.

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Step 29.3.1

Simplify each term.

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Step 29.3.1.1

Multiply $3$ by $4$ .

$\frac{12 t^{2} + 4 (3 \cdot 3) + 4 (- 2 t^{2})}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.3.1.2

Multiply $4 (3 \cdot 3)$ .

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Step 29.3.1.2.1

Multiply $3$ by $3$ .

$\frac{12 t^{2} + 4 \cdot 9 + 4 (- 2 t^{2})}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.3.1.2.2

Multiply $4$ by $9$ .

$\frac{12 t^{2} + 36 + 4 (- 2 t^{2})}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.3.1.3

Multiply $- 2$ by $4$ .

$\frac{12 t^{2} + 36 - 8 t^{2}}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.3.2

Subtract $8 t^{2}$ from $12 t^{2}$ .

$\frac{4 t^{2} + 36}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.4

Factor $4$ out of $4 t^{2} + 36$ .

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Step 29.4.1

Factor $4$ out of $4 t^{2}$ .

$\frac{4 (t^{2}) + 36}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.4.2

Factor $4$ out of $36$ .

$\frac{4 t^{2} + 4 \cdot 9}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$

Step 29.4.3

Factor $4$ out of $4 t^{2} + 4 \cdot 9$ .

$\frac{4 (t^{2} + 9)}{3 {(t^{2} + 3)}^{\frac{4}{3}}}$