Calculus Examples

$y (t) = (\frac{9 t^{5} - 3.9 t^{4} - t^{3}}{3 t^{3}})$

Step 1

Since $\frac{1}{3}$ is constant with respect to $t$ , the derivative of $\frac{9 t^{5} - 3.9 t^{4} - t^{3}}{3 t^{3}}$ with respect to $t$ is $\frac{1}{3} \frac{d}{d t} [\frac{9 t^{5} - 3.9 t^{4} - t^{3}}{t^{3}}]$ .

$\frac{1}{3} \frac{d}{d t} [\frac{9 t^{5} - 3.9 t^{4} - t^{3}}{t^{3}}]$

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 $f (t) = 9 t^{5} - 3.9 t^{4} - t^{3}$ and $g (t) = t^{3}$ .

$\frac{1}{3} \cdot \frac{t^{3} \frac{d}{d t} [9 t^{5} - 3.9 t^{4} - t^{3}] - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{{(t^{3})}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

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Step 3.1.1

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

$\frac{1}{3} \cdot \frac{t^{3} \frac{d}{d t} [9 t^{5} - 3.9 t^{4} - t^{3}] - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{3 \cdot 2}}$

Step 3.1.2

Multiply $3$ by $2$ .

$\frac{1}{3} \cdot \frac{t^{3} \frac{d}{d t} [9 t^{5} - 3.9 t^{4} - t^{3}] - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.2

By the Sum Rule, the derivative of $9 t^{5} - 3.9 t^{4} - t^{3}$ with respect to $t$ is $\frac{d}{d t} [9 t^{5}] + \frac{d}{d t} [- 3.9 t^{4}] + \frac{d}{d t} [- t^{3}]$ .

$\frac{1}{3} \cdot \frac{t^{3} (\frac{d}{d t} [9 t^{5}] + \frac{d}{d t} [- 3.9 t^{4}] + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.3

Since $9$ is constant with respect to $t$ , the derivative of $9 t^{5}$ with respect to $t$ is $9 \frac{d}{d t} [t^{5}]$ .

$\frac{1}{3} \cdot \frac{t^{3} (9 \frac{d}{d t} [t^{5}] + \frac{d}{d t} [- 3.9 t^{4}] + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.4

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

$\frac{1}{3} \cdot \frac{t^{3} (9 (5 t^{4}) + \frac{d}{d t} [- 3.9 t^{4}] + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.5

Multiply $5$ by $9$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} + \frac{d}{d t} [- 3.9 t^{4}] + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.6

Since $- 3.9$ is constant with respect to $t$ , the derivative of $- 3.9 t^{4}$ with respect to $t$ is $- 3.9 \frac{d}{d t} [t^{4}]$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 3.9 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.7

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

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 3.9 (4 t^{3}) + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.8

Multiply $4$ by $- 3.9$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} + \frac{d}{d t} [- t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.9

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{3}$ with respect to $t$ is $- \frac{d}{d t} [t^{3}]$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} - \frac{d}{d t} [t^{3}]) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.10

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

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} - (3 t^{2})) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.11

Multiply $3$ by $- 1$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - (9 t^{5} - 3.9 t^{4} - t^{3}) \frac{d}{d t} [t^{3}]}{t^{6}}$

Step 3.12

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

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - (9 t^{5} - 3.9 t^{4} - t^{3}) (3 t^{2})}{t^{6}}$

Step 3.13

Simplify with factoring out.

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Step 3.13.1

Multiply $3$ by $- 1$ .

$\frac{1}{3} \cdot \frac{t^{3} (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{5} - 3.9 t^{4} - t^{3}) t^{2}}{t^{6}}$

Step 3.13.2

Factor $t^{2}$ out of $t^{3} (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{5} - 3.9 t^{4} - t^{3}) t^{2}$ .

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Step 3.13.2.1

Factor $t^{2}$ out of $t^{3} (45 t^{4} - 15.6 t^{3} - 3 t^{2})$ .

$\frac{1}{3} \cdot \frac{t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2})) - 3 (9 t^{5} - 3.9 t^{4} - t^{3}) t^{2}}{t^{6}}$

Step 3.13.2.2

Factor $t^{2}$ out of $- 3 (9 t^{5} - 3.9 t^{4} - t^{3}) t^{2}$ .

$\frac{1}{3} \cdot \frac{t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2})) + t^{2} (- 3 (9 t^{3} - 3.9 t^{2} - t) t^{2})}{t^{6}}$

Step 3.13.2.3

Factor $t^{2}$ out of $t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2})) + t^{2} (- 3 (9 t^{3} - 3.9 t^{2} - t) t^{2})$ .

$\frac{1}{3} \cdot \frac{t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{3} - 3.9 t^{2} - t) t^{2})}{t^{6}}$

Step 4

Cancel the common factors.

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Step 4.1

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

$\frac{1}{3} \cdot \frac{t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{3} - 3.9 t^{2} - t) t^{2})}{t^{2} t^{4}}$

Step 4.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{t^{2} (t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{3} - 3.9 t^{2} - t) t^{2})}{t^{2} t^{4}}$

Step 4.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{3} - 3.9 t^{2} - t) t^{2}}{t^{4}}$

Step 5

Factor $t$ out of $t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) - 3 (9 t^{3} - 3.9 t^{2} - t) t^{2}$ .

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Step 5.1

Factor $t$ out of $- 3 (9 t^{3} - 3.9 t^{2} - t) t^{2}$ .

$\frac{1}{3} \cdot \frac{t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) + t (- 3 (9 t^{2} - 3.9 t - 1) t^{2})}{t^{4}}$

Step 5.2

Factor $t$ out of $t (45 t^{4} - 15.6 t^{3} - 3 t^{2}) + t (- 3 (9 t^{2} - 3.9 t - 1) t^{2})$ .

$\frac{1}{3} \cdot \frac{t (45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2})}{t^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

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

$\frac{1}{3} \cdot \frac{t (45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2})}{t \cdot t^{3}}$

Step 6.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{t (45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2})}{t \cdot t^{3}}$

Step 6.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2}}{t^{3}}$

Step 7

Multiply $\frac{1}{3}$ by $\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2}}{t^{3}}$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2} - 3.9 t - 1) t^{2}}{3 t^{3}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} + (- 3 (9 t^{2}) - 3 (- 3.9 t) - 3 \cdot - 1) t^{2}}{3 t^{3}}$

Step 8.2

Apply the distributive property.

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2}) t^{2} - 3 (- 3.9 t) t^{2} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Move $t^{2}$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 (t^{2} t^{2})) - 3 (- 3.9 t) t^{2} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.1.2

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

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{2 + 2}) - 3 (- 3.9 t) t^{2} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.1.3

Add $2$ and $2$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 3 (9 t^{4}) - 3 (- 3.9 t) t^{2} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.2

Multiply $9$ by $- 3$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} - 3 (- 3.9 t) t^{2} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.3

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

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Step 8.3.1.3.1

Move $t^{2}$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} - 3 (- 3.9 (t^{2} t)) - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.3.2

Multiply $t^{2}$ by $t$ .

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Step 8.3.1.3.2.1

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

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} - 3 (- 3.9 (t^{2} t^{1})) - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.3.2.2

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

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} - 3 (- 3.9 t^{2 + 1}) - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.3.3

Add $2$ and $1$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} - 3 (- 3.9 t^{3}) - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.4

Multiply $- 3.9$ by $- 3$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} + 11.7 t^{3} - 3 \cdot - 1 t^{2}}{3 t^{3}}$

Step 8.3.1.5

Multiply $- 3$ by $- 1$ .

$\frac{45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} + 11.7 t^{3} + 3 t^{2}}{3 t^{3}}$

Step 8.3.2

Combine the opposite terms in $45 t^{4} - 15.6 t^{3} - 3 t^{2} - 27 t^{4} + 11.7 t^{3} + 3 t^{2}$ .

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Step 8.3.2.1

Add $- 3 t^{2}$ and $3 t^{2}$ .

$\frac{45 t^{4} - 15.6 t^{3} - 27 t^{4} + 11.7 t^{3} + 0}{3 t^{3}}$

Step 8.3.2.2

Add $45 t^{4} - 15.6 t^{3} - 27 t^{4} + 11.7 t^{3}$ and $0$ .

$\frac{45 t^{4} - 15.6 t^{3} - 27 t^{4} + 11.7 t^{3}}{3 t^{3}}$

Step 8.3.3

Subtract $27 t^{4}$ from $45 t^{4}$ .

$\frac{18 t^{4} - 15.6 t^{3} + 11.7 t^{3}}{3 t^{3}}$

Step 8.3.4

Add $- 15.6 t^{3}$ and $11.7 t^{3}$ .

$\frac{18 t^{4} - 3.9 t^{3}}{3 t^{3}}$

Step 8.4

Factor $0.3 t^{3}$ out of $18 t^{4} - 3.9 t^{3}$ .

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Step 8.4.1

Factor $0.3 t^{3}$ out of $18 t^{4}$ .

$\frac{0.3 t^{3} (60 t) - 3.9 t^{3}}{3 t^{3}}$

Step 8.4.2

Factor $0.3 t^{3}$ out of $- 3.9 t^{3}$ .

$\frac{0.3 t^{3} (60 t) + 0.3 t^{3} (- 13)}{3 t^{3}}$

Step 8.4.3

Factor $0.3 t^{3}$ out of $0.3 t^{3} (60 t) + 0.3 t^{3} (- 13)$ .

$\frac{0.3 t^{3} (60 t - 13)}{3 t^{3}}$

Step 8.5

Cancel the common factor of $t^{3}$ .

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Step 8.5.1

Cancel the common factor.

$\frac{0.3 t^{3} (60 t - 13)}{3 t^{3}}$

Step 8.5.2

Rewrite the expression.

$\frac{0.3 (60 t - 13)}{3}$

Step 8.6

Factor $3$ out of $3$ .

$\frac{0.3 (60 t - 13)}{3 (1)}$

Step 8.7

Separate fractions.

$\frac{0.3}{3} \cdot \frac{60 t - 13}{1}$

Step 8.8

Divide $0.3$ by $3$ .

$0.1 \frac{60 t - 13}{1}$

Step 8.9

Cancel the common factor of $0.1$ .

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Step 8.9.1

Factor $0.1$ out of $1$ .

$0.1 \frac{60 t - 13}{0.1 \cdot 10}$

Step 8.9.2

Cancel the common factor.

$0.1 \frac{60 t - 13}{0.1 \cdot 10}$

Step 8.9.3

Rewrite the expression.

$\frac{60 t - 13}{10}$