Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = \frac{t^{3}}{t^{6} + 3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{t^{3}}{t^{6} + 3}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d t} [\frac{t^{3}}{t^{6} + 3}]$

Step 1.2

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

$2 u \frac{d}{d t} [\frac{t^{3}}{t^{6} + 3}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{t^{3}}{t^{6} + 3}$ .

$2 \frac{t^{3}}{t^{6} + 3} \frac{d}{d t} [\frac{t^{3}}{t^{6} + 3}]$

Step 2

Combine $2$ and $\frac{t^{3}}{t^{6} + 3}$ .

$\frac{2 t^{3}}{t^{6} + 3} \frac{d}{d t} [\frac{t^{3}}{t^{6} + 3}]$

Step 3

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 $f (t) = t^{3}$ and $g (t) = t^{6} + 3$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{(t^{6} + 3) \frac{d}{d t} [t^{3}] - t^{3} \frac{d}{d t} [t^{6} + 3]}{{(t^{6} + 3)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{(t^{6} + 3) (3 t^{2}) - t^{3} \frac{d}{d t} [t^{6} + 3]}{{(t^{6} + 3)}^{2}}$

Step 4.2

Move $3$ to the left of $t^{6} + 3$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 \cdot (t^{6} + 3) t^{2} - t^{3} \frac{d}{d t} [t^{6} + 3]}{{(t^{6} + 3)}^{2}}$

Step 4.3

By the Sum Rule, the derivative of $t^{6} + 3$ with respect to $t$ is $\frac{d}{d t} [t^{6}] + \frac{d}{d t} [3]$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - t^{3} (\frac{d}{d t} [t^{6}] + \frac{d}{d t} [3])}{{(t^{6} + 3)}^{2}}$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 6$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - t^{3} (6 t^{5} + \frac{d}{d t} [3])}{{(t^{6} + 3)}^{2}}$

Step 4.5

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

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - t^{3} (6 t^{5} + 0)}{{(t^{6} + 3)}^{2}}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $6 t^{5}$ and $0$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - t^{3} (6 t^{5})}{{(t^{6} + 3)}^{2}}$

Step 4.6.2

Multiply $6$ by $- 1$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - 6 t^{3} t^{5}}{{(t^{6} + 3)}^{2}}$

Step 5

Multiply $t^{3}$ by $t^{5}$ by adding the exponents.

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Step 5.1

Move $t^{5}$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - 6 (t^{5} t^{3})}{{(t^{6} + 3)}^{2}}$

Step 5.2

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

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - 6 t^{5 + 3}}{{(t^{6} + 3)}^{2}}$

Step 5.3

Add $5$ and $3$ .

$\frac{2 t^{3}}{t^{6} + 3} \cdot \frac{3 (t^{6} + 3) t^{2} - 6 t^{8}}{{(t^{6} + 3)}^{2}}$

Step 6

Multiply $\frac{2 t^{3}}{t^{6} + 3}$ by $\frac{3 (t^{6} + 3) t^{2} - 6 t^{8}}{{(t^{6} + 3)}^{2}}$ .

$\frac{2 t^{3} (3 (t^{6} + 3) t^{2} - 6 t^{8})}{(t^{6} + 3) {(t^{6} + 3)}^{2}}$

Step 7

Multiply $t^{6} + 3$ by ${(t^{6} + 3)}^{2}$ by adding the exponents.

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Step 7.1

Multiply $t^{6} + 3$ by ${(t^{6} + 3)}^{2}$ .

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Step 7.1.1

Raise $t^{6} + 3$ to the power of $1$ .

$\frac{2 t^{3} (3 (t^{6} + 3) t^{2} - 6 t^{8})}{{(t^{6} + 3)}^{1} {(t^{6} + 3)}^{2}}$

Step 7.1.2

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

$\frac{2 t^{3} (3 (t^{6} + 3) t^{2} - 6 t^{8})}{{(t^{6} + 3)}^{1 + 2}}$

Step 7.2

Add $1$ and $2$ .

$\frac{2 t^{3} (3 (t^{6} + 3) t^{2} - 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{2 t^{3} ((3 t^{6} + 3 \cdot 3) t^{2} - 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.2

Apply the distributive property.

$\frac{2 t^{3} (3 t^{6} t^{2} + 3 \cdot 3 t^{2} - 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.3

Apply the distributive property.

$\frac{2 t^{3} (3 t^{6} t^{2}) + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Simplify each term.

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Step 8.4.1.1

Multiply $t^{3}$ by $t^{6}$ by adding the exponents.

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Step 8.4.1.1.1

Move $t^{6}$ .

$\frac{2 (t^{6} t^{3}) (3 t^{2}) + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.1.2

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

$\frac{2 t^{6 + 3} (3 t^{2}) + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.1.3

Add $6$ and $3$ .

$\frac{2 t^{9} (3 t^{2}) + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.2

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 3 t^{9} t^{2} + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.3

Multiply $t^{9}$ by $t^{2}$ by adding the exponents.

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Step 8.4.1.3.1

Move $t^{2}$ .

$\frac{2 \cdot 3 (t^{2} t^{9}) + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.3.2

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

$\frac{2 \cdot 3 t^{2 + 9} + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.3.3

Add $2$ and $9$ .

$\frac{2 \cdot 3 t^{11} + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.4

Multiply $2$ by $3$ .

$\frac{6 t^{11} + 2 t^{3} (3 \cdot 3 t^{2}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.5

Rewrite using the commutative property of multiplication.

$\frac{6 t^{11} + 2 \cdot 3 \cdot 3 t^{3} t^{2} + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.6

Multiply $t^{3}$ by $t^{2}$ by adding the exponents.

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Step 8.4.1.6.1

Move $t^{2}$ .

$\frac{6 t^{11} + 2 \cdot 3 \cdot 3 (t^{2} t^{3}) + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.6.2

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

$\frac{6 t^{11} + 2 \cdot 3 \cdot 3 t^{2 + 3} + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.6.3

Add $2$ and $3$ .

$\frac{6 t^{11} + 2 \cdot 3 \cdot 3 t^{5} + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.7

Multiply $2 \cdot 3 \cdot 3$ .

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Step 8.4.1.7.1

Multiply $2$ by $3$ .

$\frac{6 t^{11} + 6 \cdot 3 t^{5} + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.7.2

Multiply $6$ by $3$ .

$\frac{6 t^{11} + 18 t^{5} + 2 t^{3} (- 6 t^{8})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.8

Rewrite using the commutative property of multiplication.

$\frac{6 t^{11} + 18 t^{5} + 2 \cdot - 6 t^{3} t^{8}}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.9

Multiply $t^{3}$ by $t^{8}$ by adding the exponents.

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Step 8.4.1.9.1

Move $t^{8}$ .

$\frac{6 t^{11} + 18 t^{5} + 2 \cdot - 6 (t^{8} t^{3})}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.9.2

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

$\frac{6 t^{11} + 18 t^{5} + 2 \cdot - 6 t^{8 + 3}}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.9.3

Add $8$ and $3$ .

$\frac{6 t^{11} + 18 t^{5} + 2 \cdot - 6 t^{11}}{{(t^{6} + 3)}^{3}}$

Step 8.4.1.10

Multiply $2$ by $- 6$ .

$\frac{6 t^{11} + 18 t^{5} - 12 t^{11}}{{(t^{6} + 3)}^{3}}$

Step 8.4.2

Subtract $12 t^{11}$ from $6 t^{11}$ .

$\frac{- 6 t^{11} + 18 t^{5}}{{(t^{6} + 3)}^{3}}$

Step 8.5

Factor $6 t^{5}$ out of $- 6 t^{11} + 18 t^{5}$ .

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Step 8.5.1

Factor $6 t^{5}$ out of $- 6 t^{11}$ .

$\frac{6 t^{5} (- t^{6}) + 18 t^{5}}{{(t^{6} + 3)}^{3}}$

Step 8.5.2

Factor $6 t^{5}$ out of $18 t^{5}$ .

$\frac{6 t^{5} (- t^{6}) + 6 t^{5} (3)}{{(t^{6} + 3)}^{3}}$

Step 8.5.3

Factor $6 t^{5}$ out of $6 t^{5} (- t^{6}) + 6 t^{5} (3)$ .

$\frac{6 t^{5} (- t^{6} + 3)}{{(t^{6} + 3)}^{3}}$

Step 8.6

Factor $- 1$ out of $- t^{6}$ .

$\frac{6 t^{5} (- (t^{6}) + 3)}{{(t^{6} + 3)}^{3}}$

Step 8.7

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{6 t^{5} (- (t^{6}) - 1 (- 3))}{{(t^{6} + 3)}^{3}}$

Step 8.8

Factor $- 1$ out of $- (t^{6}) - 1 (- 3)$ .

$\frac{6 t^{5} (- (t^{6} - 3))}{{(t^{6} + 3)}^{3}}$

Step 8.9

Rewrite $- (t^{6} - 3)$ as $- 1 (t^{6} - 3)$ .

$\frac{6 t^{5} (- 1 (t^{6} - 3))}{{(t^{6} + 3)}^{3}}$

Step 8.10

Move the negative in front of the fraction.

$- \frac{(6 t^{5}) (t^{6} - 3)}{{(t^{6} + 3)}^{3}}$