Calculus Examples

$\frac{2 (18 t^{2} - 9 t + 2)}{{(1 - 4 t)}^{3}}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $\frac{2 (18 t^{2} - 9 t + 2)}{{(1 - 4 t)}^{3}}$ with respect to $t$ is $2 \frac{d}{d t} [\frac{18 t^{2} - 9 t + 2}{{(1 - 4 t)}^{3}}]$ .

$2 \frac{d}{d t} [\frac{18 t^{2} - 9 t + 2}{{(1 - 4 t)}^{3}}]$

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 $f (t) = 18 t^{2} - 9 t + 2$ and $g (t) = {(1 - 4 t)}^{3}$ .

$2 \frac{{(1 - 4 t)}^{3} \frac{d}{d t} [18 t^{2} - 9 t + 2] - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{({(1 - 4 t)}^{3})}^{2}}$

Step 3

Differentiate.

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Step 3.1

Multiply the exponents in ${({(1 - 4 t)}^{3})}^{2}$ .

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Step 3.1.1

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

$2 \frac{{(1 - 4 t)}^{3} \frac{d}{d t} [18 t^{2} - 9 t + 2] - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{3 \cdot 2}}$

Step 3.1.2

Multiply $3$ by $2$ .

$2 \frac{{(1 - 4 t)}^{3} \frac{d}{d t} [18 t^{2} - 9 t + 2] - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.2

By the Sum Rule, the derivative of $18 t^{2} - 9 t + 2$ with respect to $t$ is $\frac{d}{d t} [18 t^{2}] + \frac{d}{d t} [- 9 t] + \frac{d}{d t} [2]$ .

$2 \frac{{(1 - 4 t)}^{3} (\frac{d}{d t} [18 t^{2}] + \frac{d}{d t} [- 9 t] + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.3

Since $18$ is constant with respect to $t$ , the derivative of $18 t^{2}$ with respect to $t$ is $18 \frac{d}{d t} [t^{2}]$ .

$2 \frac{{(1 - 4 t)}^{3} (18 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 9 t] + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.4

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

$2 \frac{{(1 - 4 t)}^{3} (18 (2 t) + \frac{d}{d t} [- 9 t] + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.5

Multiply $2$ by $18$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t + \frac{d}{d t} [- 9 t] + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.6

Since $- 9$ is constant with respect to $t$ , the derivative of $- 9 t$ with respect to $t$ is $- 9 \frac{d}{d t} [t]$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9 \frac{d}{d t} [t] + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.7

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

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9 \cdot 1 + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.8

Multiply $- 9$ by $1$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9 + \frac{d}{d t} [2]) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.9

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9 + 0) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 3.10

Add $36 t - 9$ and $0$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9) - (18 t^{2} - 9 t + 2) \frac{d}{d t} [{(1 - 4 t)}^{3}]}{{(1 - 4 t)}^{6}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = 1 - 4 t$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $1 - 4 t$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9) - (18 t^{2} - 9 t + 2) (\frac{d}{d u} [u^{3}] \frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{6}}$

Step 4.2

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

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9) - (18 t^{2} - 9 t + 2) (3 u^{2} \frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{6}}$

Step 4.3

Replace all occurrences of $u$ with $1 - 4 t$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9) - (18 t^{2} - 9 t + 2) (3 {(1 - 4 t)}^{2} \frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{6}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $3$ by $- 1$ .

$2 \frac{{(1 - 4 t)}^{3} (36 t - 9) - 3 (18 t^{2} - 9 t + 2) ({(1 - 4 t)}^{2} \frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{6}}$

Step 5.2

Factor ${(1 - 4 t)}^{2}$ out of ${(1 - 4 t)}^{3} (36 t - 9) - 3 (18 t^{2} - 9 t + 2) ({(1 - 4 t)}^{2} \frac{d}{d t} [1 - 4 t])$ .

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Step 5.2.1

Factor ${(1 - 4 t)}^{2}$ out of ${(1 - 4 t)}^{3} (36 t - 9)$ .

$2 \frac{{(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9)) - 3 (18 t^{2} - 9 t + 2) ({(1 - 4 t)}^{2} \frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{6}}$

Step 5.2.2

Factor ${(1 - 4 t)}^{2}$ out of $- 3 (18 t^{2} - 9 t + 2) ({(1 - 4 t)}^{2} \frac{d}{d t} [1 - 4 t])$ .

$2 \frac{{(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9)) + {(1 - 4 t)}^{2} (- 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t]))}{{(1 - 4 t)}^{6}}$

Step 5.2.3

Factor ${(1 - 4 t)}^{2}$ out of ${(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9)) + {(1 - 4 t)}^{2} (- 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t]))$ .

$2 \frac{{(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t]))}{{(1 - 4 t)}^{6}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor ${(1 - 4 t)}^{2}$ out of ${(1 - 4 t)}^{6}$ .

$2 \frac{{(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t]))}{{(1 - 4 t)}^{2} {(1 - 4 t)}^{4}}$

Step 6.2

Cancel the common factor.

$2 \frac{{(1 - 4 t)}^{2} ((1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t]))}{{(1 - 4 t)}^{2} {(1 - 4 t)}^{4}}$

Step 6.3

Rewrite the expression.

$2 \frac{(1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1 - 4 t])}{{(1 - 4 t)}^{4}}$

Step 7

By the Sum Rule, the derivative of $1 - 4 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- 4 t]$ .

$2 \frac{(1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (\frac{d}{d t} [1] + \frac{d}{d t} [- 4 t])}{{(1 - 4 t)}^{4}}$

Step 8

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$2 \frac{(1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (0 + \frac{d}{d t} [- 4 t])}{{(1 - 4 t)}^{4}}$

Step 9

Add $0$ and $\frac{d}{d t} [- 4 t]$ .

$2 \frac{(1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) \frac{d}{d t} [- 4 t]}{{(1 - 4 t)}^{4}}$

Step 10

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

$2 \frac{(1 - 4 t) (36 t - 9) - 3 (18 t^{2} - 9 t + 2) (- 4 \frac{d}{d t} [t])}{{(1 - 4 t)}^{4}}$

Step 11

Multiply $- 4$ by $- 3$ .

$2 \frac{(1 - 4 t) (36 t - 9) + 12 (18 t^{2} - 9 t + 2) \frac{d}{d t} [t]}{{(1 - 4 t)}^{4}}$

Step 12

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

$2 \frac{(1 - 4 t) (36 t - 9) + 12 (18 t^{2} - 9 t + 2) \cdot 1}{{(1 - 4 t)}^{4}}$

Step 13

Combine fractions.

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Step 13.1

Multiply $12$ by $1$ .

$2 \frac{(1 - 4 t) (36 t - 9) + 12 (18 t^{2} - 9 t + 2)}{{(1 - 4 t)}^{4}}$

Step 13.2

Combine $2$ and $\frac{(1 - 4 t) (36 t - 9) + 12 (18 t^{2} - 9 t + 2)}{{(1 - 4 t)}^{4}}$ .

$\frac{2 ((1 - 4 t) (36 t - 9) + 12 (18 t^{2} - 9 t + 2))}{{(1 - 4 t)}^{4}}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{2 ((1 - 4 t) (36 t - 9) + 12 (18 t^{2}) + 12 (- 9 t) + 12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.2

Apply the distributive property.

$\frac{2 ((1 - 4 t) (36 t - 9)) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Simplify each term.

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Step 14.3.1.1

Expand $(1 - 4 t) (36 t - 9)$ using the FOIL Method.

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Step 14.3.1.1.1

Apply the distributive property.

$\frac{2 (1 (36 t - 9) - 4 t (36 t - 9)) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.1.2

Apply the distributive property.

$\frac{2 (1 (36 t) + 1 \cdot - 9 - 4 t (36 t - 9)) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.1.3

Apply the distributive property.

$\frac{2 (1 (36 t) + 1 \cdot - 9 - 4 t (36 t) - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2

Simplify and combine like terms.

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Step 14.3.1.2.1

Simplify each term.

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Step 14.3.1.2.1.1

Multiply $36 t$ by $1$ .

$\frac{2 (36 t + 1 \cdot - 9 - 4 t (36 t) - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.2

Multiply $- 9$ by $1$ .

$\frac{2 (36 t - 9 - 4 t (36 t) - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 (36 t - 9 - 4 \cdot 36 t \cdot t - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.4

Multiply $t$ by $t$ by adding the exponents.

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Step 14.3.1.2.1.4.1

Move $t$ .

$\frac{2 (36 t - 9 - 4 \cdot 36 (t \cdot t) - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.4.2

Multiply $t$ by $t$ .

$\frac{2 (36 t - 9 - 4 \cdot 36 t^{2} - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.5

Multiply $- 4$ by $36$ .

$\frac{2 (36 t - 9 - 144 t^{2} - 4 t \cdot - 9) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.1.6

Multiply $- 9$ by $- 4$ .

$\frac{2 (36 t - 9 - 144 t^{2} + 36 t) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.2.2

Add $36 t$ and $36 t$ .

$\frac{2 (72 t - 9 - 144 t^{2}) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.3

Apply the distributive property.

$\frac{2 (72 t) + 2 \cdot - 9 + 2 (- 144 t^{2}) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.4

Simplify.

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Step 14.3.1.4.1

Multiply $72$ by $2$ .

$\frac{144 t + 2 \cdot - 9 + 2 (- 144 t^{2}) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.4.2

Multiply $2$ by $- 9$ .

$\frac{144 t - 18 + 2 (- 144 t^{2}) + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.4.3

Multiply $- 144$ by $2$ .

$\frac{144 t - 18 - 288 t^{2} + 2 (12 (18 t^{2})) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.5

Multiply $18$ by $12$ .

$\frac{144 t - 18 - 288 t^{2} + 2 (216 t^{2}) + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.6

Multiply $216$ by $2$ .

$\frac{144 t - 18 - 288 t^{2} + 432 t^{2} + 2 (12 (- 9 t)) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.7

Multiply $- 9$ by $12$ .

$\frac{144 t - 18 - 288 t^{2} + 432 t^{2} + 2 (- 108 t) + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.8

Multiply $- 108$ by $2$ .

$\frac{144 t - 18 - 288 t^{2} + 432 t^{2} - 216 t + 2 (12 \cdot 2)}{{(1 - 4 t)}^{4}}$

Step 14.3.1.9

Multiply $2 (12 \cdot 2)$ .

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Step 14.3.1.9.1

Multiply $12$ by $2$ .

$\frac{144 t - 18 - 288 t^{2} + 432 t^{2} - 216 t + 2 \cdot 24}{{(1 - 4 t)}^{4}}$

Step 14.3.1.9.2

Multiply $2$ by $24$ .

$\frac{144 t - 18 - 288 t^{2} + 432 t^{2} - 216 t + 48}{{(1 - 4 t)}^{4}}$

Step 14.3.2

Subtract $216 t$ from $144 t$ .

$\frac{- 18 - 288 t^{2} + 432 t^{2} - 72 t + 48}{{(1 - 4 t)}^{4}}$

Step 14.3.3

Add $- 18$ and $48$ .

$\frac{- 288 t^{2} + 432 t^{2} - 72 t + 30}{{(1 - 4 t)}^{4}}$

Step 14.3.4

Add $- 288 t^{2}$ and $432 t^{2}$ .

$\frac{144 t^{2} - 72 t + 30}{{(1 - 4 t)}^{4}}$

Step 14.4

Reorder terms.

$\frac{144 t^{2} - 72 t + 30}{{(- 4 t + 1)}^{4}}$

Step 14.5

Factor $6$ out of $144 t^{2} - 72 t + 30$ .

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Step 14.5.1

Factor $6$ out of $144 t^{2}$ .

$\frac{6 (24 t^{2}) - 72 t + 30}{{(- 4 t + 1)}^{4}}$

Step 14.5.2

Factor $6$ out of $- 72 t$ .

$\frac{6 (24 t^{2}) + 6 (- 12 t) + 30}{{(- 4 t + 1)}^{4}}$

Step 14.5.3

Factor $6$ out of $30$ .

$\frac{6 (24 t^{2}) + 6 (- 12 t) + 6 (5)}{{(- 4 t + 1)}^{4}}$

Step 14.5.4

Factor $6$ out of $6 (24 t^{2}) + 6 (- 12 t)$ .

$\frac{6 (24 t^{2} - 12 t) + 6 (5)}{{(- 4 t + 1)}^{4}}$

Step 14.5.5

Factor $6$ out of $6 (24 t^{2} - 12 t) + 6 (5)$ .

$\frac{6 (24 t^{2} - 12 t + 5)}{{(- 4 t + 1)}^{4}}$