Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = \frac{t^{2}}{t^{3} - 4 t}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{t^{2}}{t^{3} - 4 t}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{t^{2}}{t^{3} - 4 t}$ .

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

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 $f (t) = t^{2}$ and $g (t) = t^{3} - 4 t$ .

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

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 $n = 2$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine fractions.

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Step 3.7.1

Multiply $- 4$ by $1$ .

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{t^{2}}{t^{3} - 4 t}$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Combine terms.

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

Add $1$ and $3$ .

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

Step 4.6.4

Multiply $2$ by $3$ .

$\frac{6 t^{4} + 3 (2 (- 4 t) t) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.5

Multiply $- 4$ by $2$ .

$\frac{6 t^{4} + 3 (- 8 t \cdot t) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.6

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

$\frac{6 t^{4} + 3 (- 8 (t^{1} t)) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.7

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

$\frac{6 t^{4} + 3 (- 8 (t^{1} t^{1})) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.8

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

$\frac{6 t^{4} + 3 (- 8 t^{1 + 1}) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.9

Add $1$ and $1$ .

$\frac{6 t^{4} + 3 (- 8 t^{2}) + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.10

Multiply $- 8$ by $3$ .

$\frac{6 t^{4} - 24 t^{2} + 3 (- t^{2} (3 t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.11

Multiply $3$ by $- 1$ .

$\frac{6 t^{4} - 24 t^{2} + 3 (- 3 t^{2} t^{2}) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.12

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

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Step 4.6.12.1

Move $t^{2}$ .

$\frac{6 t^{4} - 24 t^{2} + 3 (- 3 (t^{2} t^{2})) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.12.2

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

$\frac{6 t^{4} - 24 t^{2} + 3 (- 3 t^{2 + 2}) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.12.3

Add $2$ and $2$ .

$\frac{6 t^{4} - 24 t^{2} + 3 (- 3 t^{4}) + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.13

Multiply $- 3$ by $3$ .

$\frac{6 t^{4} - 24 t^{2} - 9 t^{4} + 3 (- t^{2} \cdot - 4)}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.14

Multiply $- 4$ by $- 1$ .

$\frac{6 t^{4} - 24 t^{2} - 9 t^{4} + 3 (4 t^{2})}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.15

Multiply $4$ by $3$ .

$\frac{6 t^{4} - 24 t^{2} - 9 t^{4} + 12 t^{2}}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.16

Subtract $9 t^{4}$ from $6 t^{4}$ .

$\frac{- 3 t^{4} - 24 t^{2} + 12 t^{2}}{{(t^{3} - 4 t)}^{2}} \cdot \frac{{(t^{2})}^{2}}{{(t^{3} - 4 t)}^{2}}$

Step 4.6.17

Add $- 24 t^{2}$ and $12 t^{2}$ .

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

Step 4.6.18

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 4.6.18.1

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

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

Step 4.6.18.2

Multiply $2$ by $2$ .

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

Step 4.6.19

Multiply $\frac{- 3 t^{4} - 12 t^{2}}{{(t^{3} - 4 t)}^{2}}$ by $\frac{t^{4}}{{(t^{3} - 4 t)}^{2}}$ .

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

Step 4.6.20

Multiply ${(t^{3} - 4 t)}^{2}$ by ${(t^{3} - 4 t)}^{2}$ by adding the exponents.

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Step 4.6.20.1

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

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

Step 4.6.20.2

Add $2$ and $2$ .

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

Step 4.7

Reorder terms.

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

Step 4.8

Simplify the numerator.

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Step 4.8.1

Factor $3 t^{2}$ out of $- 3 t^{4} - 12 t^{2}$ .

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Step 4.8.1.1

Factor $3 t^{2}$ out of $- 3 t^{4}$ .

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

Step 4.8.1.2

Factor $3 t^{2}$ out of $- 12 t^{2}$ .

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

Step 4.8.1.3

Factor $3 t^{2}$ out of $3 t^{2} (- t^{2}) + 3 t^{2} (- 4)$ .

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

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

Step 4.8.2

Multiply $t^{4}$ by $t^{2}$ by adding the exponents.

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Step 4.8.2.1

Move $t^{2}$ .

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

Step 4.8.2.2

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

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

Step 4.8.2.3

Add $2$ and $4$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t^{3} - 4 t)}^{4}}$

Step 4.9

Simplify the denominator.

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Step 4.9.1

Factor $t$ out of $t^{3} - 4 t$ .

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Step 4.9.1.1

Factor $t$ out of $t^{3}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t \cdot t^{2} - 4 t)}^{4}}$

Step 4.9.1.2

Factor $t$ out of $- 4 t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t \cdot t^{2} + t \cdot - 4)}^{4}}$

Step 4.9.1.3

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t (t^{2} - 4))}^{4}}$

Step 4.9.2

Rewrite $4$ as $2^{2}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t (t^{2} - 2^{2}))}^{4}}$

Step 4.9.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 2$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t (t + 2) (t - 2))}^{4}}$

Step 4.9.4

Apply the product rule to $t (t + 2) (t - 2)$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t (t + 2))}^{4} {(t - 2)}^{4}}$

Step 4.9.5

Apply the distributive property.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t \cdot t + t \cdot 2)}^{4} {(t - 2)}^{4}}$

Step 4.9.6

Multiply $t$ by $t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t^{2} + t \cdot 2)}^{4} {(t - 2)}^{4}}$

Step 4.9.7

Move $2$ to the left of $t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{{(t^{2} + 2 \cdot t)}^{4} {(t - 2)}^{4}}$

Step 4.9.8

Use the Binomial Theorem.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{({(t^{2})}^{4} + 4 {(t^{2})}^{3} (2 t) + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9

Simplify each term.

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Step 4.9.9.1

Multiply the exponents in ${(t^{2})}^{4}$ .

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Step 4.9.9.1.1

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{2 \cdot 4} + 4 {(t^{2})}^{3} (2 t) + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.1.2

Multiply $2$ by $4$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 4 {(t^{2})}^{3} (2 t) + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.2

Rewrite using the commutative property of multiplication.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 4 \cdot 2 {(t^{2})}^{3} t + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.3

Multiply $4$ by $2$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 {(t^{2})}^{3} t + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.4

Multiply the exponents in ${(t^{2})}^{3}$ .

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Step 4.9.9.4.1

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{2 \cdot 3} t + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.4.2

Multiply $2$ by $3$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{6} t + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.5

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

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Step 4.9.9.5.1

Move $t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 (t \cdot t^{6}) + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.5.2

Multiply $t$ by $t^{6}$ .

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Step 4.9.9.5.2.1

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 (t^{1} t^{6}) + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.5.2.2

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{1 + 6} + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.5.3

Add $1$ and $6$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 {(t^{2})}^{2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.6

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 4.9.9.6.1

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 t^{2 \cdot 2} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.6.2

Multiply $2$ by $2$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 t^{4} {(2 t)}^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.7

Apply the product rule to $2 t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 t^{4} (2^{2} t^{2}) + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.8

Rewrite using the commutative property of multiplication.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 \cdot 2^{2} t^{4} t^{2} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.9

Multiply $t^{4}$ by $t^{2}$ by adding the exponents.

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Step 4.9.9.9.1

Move $t^{2}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 \cdot 2^{2} (t^{2} t^{4}) + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.9.2

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 \cdot 2^{2} t^{2 + 4} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.9.3

Add $2$ and $4$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 \cdot 2^{2} t^{6} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.10

Raise $2$ to the power of $2$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 6 \cdot 4 t^{6} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.11

Multiply $6$ by $4$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 t^{2} {(2 t)}^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.12

Apply the product rule to $2 t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 t^{2} (2^{3} t^{3}) + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.13

Rewrite using the commutative property of multiplication.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 \cdot 2^{3} t^{2} t^{3} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.14

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

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Step 4.9.9.14.1

Move $t^{3}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 \cdot 2^{3} (t^{3} t^{2}) + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.14.2

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

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 \cdot 2^{3} t^{3 + 2} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.14.3

Add $3$ and $2$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 \cdot 2^{3} t^{5} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.15

Raise $2$ to the power of $3$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 4 \cdot 8 t^{5} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.16

Multiply $4$ by $8$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 32 t^{5} + {(2 t)}^{4}) {(t - 2)}^{4}}$

Step 4.9.9.17

Apply the product rule to $2 t$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 32 t^{5} + 2^{4} t^{4}) {(t - 2)}^{4}}$

Step 4.9.9.18

Raise $2$ to the power of $4$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{8} + 8 t^{7} + 24 t^{6} + 32 t^{5} + 16 t^{4}) {(t - 2)}^{4}}$

Step 4.9.10

Factor $t^{4}$ out of $t^{8} + 8 t^{7} + 24 t^{6} + 32 t^{5} + 16 t^{4}$ .

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Step 4.9.10.1

Factor $t^{4}$ out of $t^{8}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} t^{4} + 8 t^{7} + 24 t^{6} + 32 t^{5} + 16 t^{4}) {(t - 2)}^{4}}$

Step 4.9.10.2

Factor $t^{4}$ out of $8 t^{7}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} t^{4} + t^{4} (8 t^{3}) + 24 t^{6} + 32 t^{5} + 16 t^{4}) {(t - 2)}^{4}}$

Step 4.9.10.3

Factor $t^{4}$ out of $24 t^{6}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} t^{4} + t^{4} (8 t^{3}) + t^{4} (24 t^{2}) + 32 t^{5} + 16 t^{4}) {(t - 2)}^{4}}$

Step 4.9.10.4

Factor $t^{4}$ out of $32 t^{5}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} t^{4} + t^{4} (8 t^{3}) + t^{4} (24 t^{2}) + t^{4} (32 t) + 16 t^{4}) {(t - 2)}^{4}}$

Step 4.9.10.5

Factor $t^{4}$ out of $16 t^{4}$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} t^{4} + t^{4} (8 t^{3}) + t^{4} (24 t^{2}) + t^{4} (32 t) + t^{4} \cdot 16) {(t - 2)}^{4}}$

Step 4.9.10.6

Factor $t^{4}$ out of $t^{4} t^{4} + t^{4} (8 t^{3})$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} (t^{4} + 8 t^{3}) + t^{4} (24 t^{2}) + t^{4} (32 t) + t^{4} \cdot 16) {(t - 2)}^{4}}$

Step 4.9.10.7

Factor $t^{4}$ out of $t^{4} (t^{4} + 8 t^{3}) + t^{4} (24 t^{2})$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} (t^{4} + 8 t^{3} + 24 t^{2}) + t^{4} (32 t) + t^{4} \cdot 16) {(t - 2)}^{4}}$

Step 4.9.10.8

Factor $t^{4}$ out of $t^{4} (t^{4} + 8 t^{3} + 24 t^{2}) + t^{4} (32 t)$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{(t^{4} (t^{4} + 8 t^{3} + 24 t^{2} + 32 t) + t^{4} \cdot 16) {(t - 2)}^{4}}$

Step 4.9.10.9

Factor $t^{4}$ out of $t^{4} (t^{4} + 8 t^{3} + 24 t^{2} + 32 t) + t^{4} \cdot 16$ .

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{t^{4} (t^{4} + 8 t^{3} + 24 t^{2} + 32 t + 16) {(t - 2)}^{4}}$

Step 4.9.11

Make each term match the terms from the binomial theorem formula.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{t^{4} (t^{4} + 4 \cdot 2 t^{3} + 6 \cdot 2^{2} t^{2} + 4 \cdot 2^{3} t + 2^{4}) {(t - 2)}^{4}}$

Step 4.9.12

Factor $t^{4} + 4 \cdot 2 t^{3} + 6 \cdot 2^{2} t^{2} + 4 \cdot 2^{3} t + 2^{4}$ using the binomial theorem.

$\frac{t^{6} \cdot 3 (- t^{2} - 4)}{t^{4} {(t + 2)}^{4} {(t - 2)}^{4}}$

Step 4.10

Cancel the common factor of $t^{6}$ and $t^{4}$ .

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Step 4.10.1

Factor $t^{4}$ out of $t^{6} \cdot 3 (- t^{2} - 4)$ .

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

Step 4.10.2

Cancel the common factors.

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Step 4.10.2.1

Factor $t^{4}$ out of $t^{4} {(t + 2)}^{4} {(t - 2)}^{4}$ .

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

Step 4.10.2.2

Cancel the common factor.

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

Step 4.10.2.3

Rewrite the expression.

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.14

Factor $- 1$ out of $- (t^{2}) - 1 (4)$ .

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

Step 4.15

Rewrite $- (t^{2} + 4)$ as $- 1 (t^{2} + 4)$ .

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

Step 4.16

Move the negative in front of the fraction.

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

Step 4.17

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

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