Calculus Examples

$f (t) = (1.3 t^{7} + 9.1 t^{2}) (2.1 t^{8} + 1.8 t^{5})$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 1.3 t^{7} + 9.1 t^{2}$ and $g (t) = 2.1 t^{8} + 1.8 t^{5}$ .

$(1.3 t^{7} + 9.1 t^{2}) \frac{d}{d t} [2.1 t^{8} + 1.8 t^{5}] + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2.1 t^{8} + 1.8 t^{5}$ with respect to $t$ is $\frac{d}{d t} [2.1 t^{8}] + \frac{d}{d t} [1.8 t^{5}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (\frac{d}{d t} [2.1 t^{8}] + \frac{d}{d t} [1.8 t^{5}]) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.2

Since $2.1$ is constant with respect to $t$ , the derivative of $2.1 t^{8}$ with respect to $t$ is $2.1 \frac{d}{d t} [t^{8}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (2.1 \frac{d}{d t} [t^{8}] + \frac{d}{d t} [1.8 t^{5}]) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 8$ .

$(1.3 t^{7} + 9.1 t^{2}) (2.1 (8 t^{7}) + \frac{d}{d t} [1.8 t^{5}]) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.4

Multiply $8$ by $2.1$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + \frac{d}{d t} [1.8 t^{5}]) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.5

Since $1.8$ is constant with respect to $t$ , the derivative of $1.8 t^{5}$ with respect to $t$ is $1.8 \frac{d}{d t} [t^{5}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 1.8 \frac{d}{d t} [t^{5}]) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.6

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 1.8 (5 t^{4})) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.7

Multiply $5$ by $1.8$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) \frac{d}{d t} [1.3 t^{7} + 9.1 t^{2}]$

Step 1.2.8

By the Sum Rule, the derivative of $1.3 t^{7} + 9.1 t^{2}$ with respect to $t$ is $\frac{d}{d t} [1.3 t^{7}] + \frac{d}{d t} [9.1 t^{2}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (\frac{d}{d t} [1.3 t^{7}] + \frac{d}{d t} [9.1 t^{2}])$

Step 1.2.9

Since $1.3$ is constant with respect to $t$ , the derivative of $1.3 t^{7}$ with respect to $t$ is $1.3 \frac{d}{d t} [t^{7}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (1.3 \frac{d}{d t} [t^{7}] + \frac{d}{d t} [9.1 t^{2}])$

Step 1.2.10

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (1.3 (7 t^{6}) + \frac{d}{d t} [9.1 t^{2}])$

Step 1.2.11

Multiply $7$ by $1.3$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + \frac{d}{d t} [9.1 t^{2}])$

Step 1.2.12

Since $9.1$ is constant with respect to $t$ , the derivative of $9.1 t^{2}$ with respect to $t$ is $9.1 \frac{d}{d t} [t^{2}]$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 9.1 \frac{d}{d t} [t^{2}])$

Step 1.2.13

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

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 9.1 (2 t))$

Step 1.2.14

Multiply $2$ by $9.1$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3

Simplify.

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Step 1.3.1

Factor $1$ out of $(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$ .

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Step 1.3.1.1

Factor $1$ out of $(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4})$ .

$1 ((1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4})) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.1.2

Factor $1$ out of $(2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$ .

$1 ((1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4})) + 1 ((2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t))$

Step 1.3.1.3

Factor $1$ out of $1 ((1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4})) + 1 ((2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t))$ .

$1 ((1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t))$

Step 1.3.2

Multiply $(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$ by $1$ .

$(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3

Simplify each term.

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Step 1.3.3.1

Expand $(1.3 t^{7} + 9.1 t^{2}) (16.8 t^{7} + 9 t^{4})$ using the FOIL Method.

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Step 1.3.3.1.1

Apply the distributive property.

$1.3 t^{7} (16.8 t^{7} + 9 t^{4}) + 9.1 t^{2} (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.1.2

Apply the distributive property.

$1.3 t^{7} (16.8 t^{7}) + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7} + 9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.1.3

Apply the distributive property.

$1.3 t^{7} (16.8 t^{7}) + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2

Simplify each term.

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Step 1.3.3.2.1

Rewrite using the commutative property of multiplication.

$1.3 \cdot 16.8 t^{7} t^{7} + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.2

Multiply $t^{7}$ by $t^{7}$ by adding the exponents.

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Step 1.3.3.2.2.1

Move $t^{7}$ .

$1.3 \cdot 16.8 (t^{7} t^{7}) + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.2.2

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

$1.3 \cdot 16.8 t^{7 + 7} + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.2.3

Add $7$ and $7$ .

$1.3 \cdot 16.8 t^{14} + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.3

Multiply $1.3$ by $16.8$ .

$21.84 t^{14} + 1.3 t^{7} (9 t^{4}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.4

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 1.3 \cdot 9 t^{7} t^{4} + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.5

Multiply $t^{7}$ by $t^{4}$ by adding the exponents.

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Step 1.3.3.2.5.1

Move $t^{4}$ .

$21.84 t^{14} + 1.3 \cdot 9 (t^{4} t^{7}) + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.5.2

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

$21.84 t^{14} + 1.3 \cdot 9 t^{4 + 7} + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.5.3

Add $4$ and $7$ .

$21.84 t^{14} + 1.3 \cdot 9 t^{11} + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.6

Multiply $1.3$ by $9$ .

$21.84 t^{14} + 11.7 t^{11} + 9.1 t^{2} (16.8 t^{7}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.7

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 9.1 \cdot 16.8 t^{2} t^{7} + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.8

Multiply $t^{2}$ by $t^{7}$ by adding the exponents.

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Step 1.3.3.2.8.1

Move $t^{7}$ .

$21.84 t^{14} + 11.7 t^{11} + 9.1 \cdot 16.8 (t^{7} t^{2}) + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.8.2

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

$21.84 t^{14} + 11.7 t^{11} + 9.1 \cdot 16.8 t^{7 + 2} + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.8.3

Add $7$ and $2$ .

$21.84 t^{14} + 11.7 t^{11} + 9.1 \cdot 16.8 t^{9} + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.9

Multiply $9.1$ by $16.8$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 9.1 t^{2} (9 t^{4}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.10

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 9.1 \cdot 9 t^{2} t^{4} + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.11

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

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Step 1.3.3.2.11.1

Move $t^{4}$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 9.1 \cdot 9 (t^{4} t^{2}) + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.11.2

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 9.1 \cdot 9 t^{4 + 2} + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.11.3

Add $4$ and $2$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 9.1 \cdot 9 t^{6} + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.2.12

Multiply $9.1$ by $9$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + (2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$

Step 1.3.3.3

Expand $(2.1 t^{8} + 1.8 t^{5}) (9.1 t^{6} + 18.2 t)$ using the FOIL Method.

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Step 1.3.3.3.1

Apply the distributive property.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 t^{8} (9.1 t^{6} + 18.2 t) + 1.8 t^{5} (9.1 t^{6} + 18.2 t)$

Step 1.3.3.3.2

Apply the distributive property.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 t^{8} (9.1 t^{6}) + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6} + 18.2 t)$

Step 1.3.3.3.3

Apply the distributive property.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 t^{8} (9.1 t^{6}) + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4

Simplify each term.

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Step 1.3.3.4.1

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 \cdot 9.1 t^{8} t^{6} + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.2

Multiply $t^{8}$ by $t^{6}$ by adding the exponents.

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Step 1.3.3.4.2.1

Move $t^{6}$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 \cdot 9.1 (t^{6} t^{8}) + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.2.2

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 \cdot 9.1 t^{6 + 8} + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.2.3

Add $6$ and $8$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 2.1 \cdot 9.1 t^{14} + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.3

Multiply $2.1$ by $9.1$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 t^{8} (18.2 t) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.4

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 \cdot 18.2 t^{8} t + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.5

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

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Step 1.3.3.4.5.1

Move $t$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 \cdot 18.2 (t \cdot t^{8}) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.5.2

Multiply $t$ by $t^{8}$ .

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Step 1.3.3.4.5.2.1

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 \cdot 18.2 (t^{1} t^{8}) + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.5.2.2

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 \cdot 18.2 t^{1 + 8} + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.5.3

Add $1$ and $8$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 2.1 \cdot 18.2 t^{9} + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.6

Multiply $2.1$ by $18.2$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 1.8 t^{5} (9.1 t^{6}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.7

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 1.8 \cdot 9.1 t^{5} t^{6} + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.8

Multiply $t^{5}$ by $t^{6}$ by adding the exponents.

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Step 1.3.3.4.8.1

Move $t^{6}$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 1.8 \cdot 9.1 (t^{6} t^{5}) + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.8.2

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 1.8 \cdot 9.1 t^{6 + 5} + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.8.3

Add $6$ and $5$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 1.8 \cdot 9.1 t^{11} + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.9

Multiply $1.8$ by $9.1$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 t^{5} (18.2 t)$

Step 1.3.3.4.10

Rewrite using the commutative property of multiplication.

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 \cdot 18.2 t^{5} t$

Step 1.3.3.4.11

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

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Step 1.3.3.4.11.1

Move $t$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 \cdot 18.2 (t \cdot t^{5})$

Step 1.3.3.4.11.2

Multiply $t$ by $t^{5}$ .

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Step 1.3.3.4.11.2.1

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 \cdot 18.2 (t^{1} t^{5})$

Step 1.3.3.4.11.2.2

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

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 \cdot 18.2 t^{1 + 5}$

Step 1.3.3.4.11.3

Add $1$ and $5$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 1.8 \cdot 18.2 t^{6}$

Step 1.3.3.4.12

Multiply $1.8$ by $18.2$ .

$21.84 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 19.11 t^{14} + 38.22 t^{9} + 16.38 t^{11} + 32.76 t^{6}$

Step 1.3.4

Add $21.84 t^{14}$ and $19.11 t^{14}$ .

$40.95 t^{14} + 11.7 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 38.22 t^{9} + 16.38 t^{11} + 32.76 t^{6}$

Step 1.3.5

Add $11.7 t^{11}$ and $16.38 t^{11}$ .

$40.95 t^{14} + 28.08 t^{11} + 152.88 t^{9} + 81.9 t^{6} + 38.22 t^{9} + 32.76 t^{6}$

Step 1.3.6

Add $152.88 t^{9}$ and $38.22 t^{9}$ .

$40.95 t^{14} + 28.08 t^{11} + 191.1 t^{9} + 81.9 t^{6} + 32.76 t^{6}$

Step 1.3.7

Add $81.9 t^{6}$ and $32.76 t^{6}$ .

$f' (t) = 40.95 t^{14} + 28.08 t^{11} + 191.1 t^{9} + 114.66 t^{6}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $40.95 t^{14} + 28.08 t^{11} + 191.1 t^{9} + 114.66 t^{6}$ with respect to $t$ is $\frac{d}{d t} [40.95 t^{14}] + \frac{d}{d t} [28.08 t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$ .

$\frac{d}{d t} [40.95 t^{14}] + \frac{d}{d t} [28.08 t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.2

Evaluate $\frac{d}{d t} [40.95 t^{14}]$ .

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Step 2.2.1

Since $40.95$ is constant with respect to $t$ , the derivative of $40.95 t^{14}$ with respect to $t$ is $40.95 \frac{d}{d t} [t^{14}]$ .

$40.95 \frac{d}{d t} [t^{14}] + \frac{d}{d t} [28.08 t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 14$ .

$40.95 (14 t^{13}) + \frac{d}{d t} [28.08 t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.2.3

Multiply $14$ by $40.95$ .

$573.3 t^{13} + \frac{d}{d t} [28.08 t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.3

Evaluate $\frac{d}{d t} [28.08 t^{11}]$ .

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Step 2.3.1

Since $28.08$ is constant with respect to $t$ , the derivative of $28.08 t^{11}$ with respect to $t$ is $28.08 \frac{d}{d t} [t^{11}]$ .

$573.3 t^{13} + 28.08 \frac{d}{d t} [t^{11}] + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 11$ .

$573.3 t^{13} + 28.08 (11 t^{10}) + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.3.3

Multiply $11$ by $28.08$ .

$573.3 t^{13} + 308.88 t^{10} + \frac{d}{d t} [191.1 t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.4

Evaluate $\frac{d}{d t} [191.1 t^{9}]$ .

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Step 2.4.1

Since $191.1$ is constant with respect to $t$ , the derivative of $191.1 t^{9}$ with respect to $t$ is $191.1 \frac{d}{d t} [t^{9}]$ .

$573.3 t^{13} + 308.88 t^{10} + 191.1 \frac{d}{d t} [t^{9}] + \frac{d}{d t} [114.66 t^{6}]$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 9$ .

$573.3 t^{13} + 308.88 t^{10} + 191.1 (9 t^{8}) + \frac{d}{d t} [114.66 t^{6}]$

Step 2.4.3

Multiply $9$ by $191.1$ .

$573.3 t^{13} + 308.88 t^{10} + 1719.9 t^{8} + \frac{d}{d t} [114.66 t^{6}]$

Step 2.5

Evaluate $\frac{d}{d t} [114.66 t^{6}]$ .

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Step 2.5.1

Since $114.66$ is constant with respect to $t$ , the derivative of $114.66 t^{6}$ with respect to $t$ is $114.66 \frac{d}{d t} [t^{6}]$ .

$573.3 t^{13} + 308.88 t^{10} + 1719.9 t^{8} + 114.66 \frac{d}{d t} [t^{6}]$

Step 2.5.2

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

$573.3 t^{13} + 308.88 t^{10} + 1719.9 t^{8} + 114.66 (6 t^{5})$

Step 2.5.3

Multiply $6$ by $114.66$ .

$f'' (t) = 573.3 t^{13} + 308.88 t^{10} + 1719.9 t^{8} + 687.96 t^{5}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $573.3 t^{13} + 308.88 t^{10} + 1719.9 t^{8} + 687.96 t^{5}$ with respect to $t$ is $\frac{d}{d t} [573.3 t^{13}] + \frac{d}{d t} [308.88 t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$ .

$\frac{d}{d t} [573.3 t^{13}] + \frac{d}{d t} [308.88 t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.2

Evaluate $\frac{d}{d t} [573.3 t^{13}]$ .

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Step 3.2.1

Since $573.3$ is constant with respect to $t$ , the derivative of $573.3 t^{13}$ with respect to $t$ is $573.3 \frac{d}{d t} [t^{13}]$ .

$573.3 \frac{d}{d t} [t^{13}] + \frac{d}{d t} [308.88 t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 13$ .

$573.3 (13 t^{12}) + \frac{d}{d t} [308.88 t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.2.3

Multiply $13$ by $573.3$ .

$7452.9 t^{12} + \frac{d}{d t} [308.88 t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.3

Evaluate $\frac{d}{d t} [308.88 t^{10}]$ .

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Step 3.3.1

Since $308.88$ is constant with respect to $t$ , the derivative of $308.88 t^{10}$ with respect to $t$ is $308.88 \frac{d}{d t} [t^{10}]$ .

$7452.9 t^{12} + 308.88 \frac{d}{d t} [t^{10}] + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 10$ .

$7452.9 t^{12} + 308.88 (10 t^{9}) + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.3.3

Multiply $10$ by $308.88$ .

$7452.9 t^{12} + 3088.8 t^{9} + \frac{d}{d t} [1719.9 t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.4

Evaluate $\frac{d}{d t} [1719.9 t^{8}]$ .

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Step 3.4.1

Since $1719.9$ is constant with respect to $t$ , the derivative of $1719.9 t^{8}$ with respect to $t$ is $1719.9 \frac{d}{d t} [t^{8}]$ .

$7452.9 t^{12} + 3088.8 t^{9} + 1719.9 \frac{d}{d t} [t^{8}] + \frac{d}{d t} [687.96 t^{5}]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 8$ .

$7452.9 t^{12} + 3088.8 t^{9} + 1719.9 (8 t^{7}) + \frac{d}{d t} [687.96 t^{5}]$

Step 3.4.3

Multiply $8$ by $1719.9$ .

$7452.9 t^{12} + 3088.8 t^{9} + 13759.2 t^{7} + \frac{d}{d t} [687.96 t^{5}]$

Step 3.5

Evaluate $\frac{d}{d t} [687.96 t^{5}]$ .

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Step 3.5.1

Since $687.96$ is constant with respect to $t$ , the derivative of $687.96 t^{5}$ with respect to $t$ is $687.96 \frac{d}{d t} [t^{5}]$ .

$7452.9 t^{12} + 3088.8 t^{9} + 13759.2 t^{7} + 687.96 \frac{d}{d t} [t^{5}]$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$7452.9 t^{12} + 3088.8 t^{9} + 13759.2 t^{7} + 687.96 (5 t^{4})$

Step 3.5.3

Multiply $5$ by $687.96$ .

$f''' (t) = 7452.9 t^{12} + 3088.8 t^{9} + 13759.2 t^{7} + 3439.8 t^{4}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $7452.9 t^{12} + 3088.8 t^{9} + 13759.2 t^{7} + 3439.8 t^{4}$ with respect to $t$ is $\frac{d}{d t} [7452.9 t^{12}] + \frac{d}{d t} [3088.8 t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$ .

$\frac{d}{d t} [7452.9 t^{12}] + \frac{d}{d t} [3088.8 t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.2

Evaluate $\frac{d}{d t} [7452.9 t^{12}]$ .

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Step 4.2.1

Since $7452.9$ is constant with respect to $t$ , the derivative of $7452.9 t^{12}$ with respect to $t$ is $7452.9 \frac{d}{d t} [t^{12}]$ .

$7452.9 \frac{d}{d t} [t^{12}] + \frac{d}{d t} [3088.8 t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 12$ .

$7452.9 (12 t^{11}) + \frac{d}{d t} [3088.8 t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.2.3

Multiply $12$ by $7452.9$ .

$89434.8 t^{11} + \frac{d}{d t} [3088.8 t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.3

Evaluate $\frac{d}{d t} [3088.8 t^{9}]$ .

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Step 4.3.1

Since $3088.8$ is constant with respect to $t$ , the derivative of $3088.8 t^{9}$ with respect to $t$ is $3088.8 \frac{d}{d t} [t^{9}]$ .

$89434.8 t^{11} + 3088.8 \frac{d}{d t} [t^{9}] + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 9$ .

$89434.8 t^{11} + 3088.8 (9 t^{8}) + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.3.3

Multiply $9$ by $3088.8$ .

$89434.8 t^{11} + 27799.2 t^{8} + \frac{d}{d t} [13759.2 t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.4

Evaluate $\frac{d}{d t} [13759.2 t^{7}]$ .

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Step 4.4.1

Since $13759.2$ is constant with respect to $t$ , the derivative of $13759.2 t^{7}$ with respect to $t$ is $13759.2 \frac{d}{d t} [t^{7}]$ .

$89434.8 t^{11} + 27799.2 t^{8} + 13759.2 \frac{d}{d t} [t^{7}] + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.4.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$89434.8 t^{11} + 27799.2 t^{8} + 13759.2 (7 t^{6}) + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.4.3

Multiply $7$ by $13759.2$ .

$89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + \frac{d}{d t} [3439.8 t^{4}]$

Step 4.5

Evaluate $\frac{d}{d t} [3439.8 t^{4}]$ .

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Step 4.5.1

Since $3439.8$ is constant with respect to $t$ , the derivative of $3439.8 t^{4}$ with respect to $t$ is $3439.8 \frac{d}{d t} [t^{4}]$ .

$89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + 3439.8 \frac{d}{d t} [t^{4}]$

Step 4.5.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + 3439.8 (4 t^{3})$

Step 4.5.3

Multiply $4$ by $3439.8$ .

$f^{4} (t) = 89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + 13759.2 t^{3}$

Step 5

The fourth derivative of $f (t)$ with respect to $t$ is $89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + 13759.2 t^{3}$ .

$89434.8 t^{11} + 27799.2 t^{8} + 96314.4 t^{6} + 13759.2 t^{3}$