Calculus Examples

$m (t) = - 5 t {(6 t^{5} - 1)}^{3}$

Step 1

Since $- 5$ is constant with respect to $t$ , the derivative of $- 5 t {(6 t^{5} - 1)}^{3}$ with respect to $t$ is $- 5 \frac{d}{d t} [t {(6 t^{5} - 1)}^{3}]$ .

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

Step 2

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 $f (t) = t$ and $g (t) = {(6 t^{5} - 1)}^{3}$ .

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

Step 3

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 $g (t) = 6 t^{5} - 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $6 t^{5} - 1$ .

$- 5 (t (\frac{d}{d u} [u^{3}] \frac{d}{d t} [6 t^{5} - 1]) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 3.2

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

$- 5 (t (3 u^{2} \frac{d}{d t} [6 t^{5} - 1]) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 3.3

Replace all occurrences of $u$ with $6 t^{5} - 1$ .

$- 5 (t (3 {(6 t^{5} - 1)}^{2} \frac{d}{d t} [6 t^{5} - 1]) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $6 t^{5} - 1$ with respect to $t$ is $\frac{d}{d t} [6 t^{5}] + \frac{d}{d t} [- 1]$ .

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (\frac{d}{d t} [6 t^{5}] + \frac{d}{d t} [- 1])) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.2

Since $6$ is constant with respect to $t$ , the derivative of $6 t^{5}$ with respect to $t$ is $6 \frac{d}{d t} [t^{5}]$ .

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (6 \frac{d}{d t} [t^{5}] + \frac{d}{d t} [- 1])) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.3

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

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (6 (5 t^{4}) + \frac{d}{d t} [- 1])) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.4

Multiply $5$ by $6$ .

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (30 t^{4} + \frac{d}{d t} [- 1])) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.5

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

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (30 t^{4} + 0)) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $30 t^{4}$ and $0$ .

$- 5 (t (3 {(6 t^{5} - 1)}^{2} (30 t^{4})) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 4.6.2

Multiply $30$ by $3$ .

$- 5 (t (90 {(6 t^{5} - 1)}^{2} t^{4}) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 5

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

$- 5 (t^{4} t^{1} (90 {(6 t^{5} - 1)}^{2}) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 6

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

$- 5 (t^{4 + 1} (90 {(6 t^{5} - 1)}^{2}) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 7

Add $4$ and $1$ .

$- 5 (t^{5} (90 {(6 t^{5} - 1)}^{2}) + {(6 t^{5} - 1)}^{3} \frac{d}{d t} [t])$

Step 8

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

$- 5 (t^{5} (90 {(6 t^{5} - 1)}^{2}) + {(6 t^{5} - 1)}^{3} \cdot 1)$

Step 9

Multiply ${(6 t^{5} - 1)}^{3}$ by $1$ .

$- 5 (t^{5} (90 {(6 t^{5} - 1)}^{2}) + {(6 t^{5} - 1)}^{3})$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- 5 (t^{5} (90 {(6 t^{5} - 1)}^{2})) - 5 {(6 t^{5} - 1)}^{3}$

Step 10.2

Multiply $90$ by $- 5$ .

$- 450 (t^{5} ({(6 t^{5} - 1)}^{2})) - 5 {(6 t^{5} - 1)}^{3}$

Step 10.3

Factor $5 {(6 t^{5} - 1)}^{2}$ out of $- 450 t^{5} {(6 t^{5} - 1)}^{2} - 5 {(6 t^{5} - 1)}^{3}$ .

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Step 10.3.1

Factor $5 {(6 t^{5} - 1)}^{2}$ out of $- 450 t^{5} {(6 t^{5} - 1)}^{2}$ .

$5 {(6 t^{5} - 1)}^{2} (- 90 t^{5}) - 5 {(6 t^{5} - 1)}^{3}$

Step 10.3.2

Factor $5 {(6 t^{5} - 1)}^{2}$ out of $- 5 {(6 t^{5} - 1)}^{3}$ .

$5 {(6 t^{5} - 1)}^{2} (- 90 t^{5}) + 5 {(6 t^{5} - 1)}^{2} (- (6 t^{5} - 1))$

Step 10.3.3

Factor $5 {(6 t^{5} - 1)}^{2}$ out of $5 {(6 t^{5} - 1)}^{2} (- 90 t^{5}) + 5 {(6 t^{5} - 1)}^{2} (- (6 t^{5} - 1))$ .

$5 {(6 t^{5} - 1)}^{2} (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.4

Rewrite ${(6 t^{5} - 1)}^{2}$ as $(6 t^{5} - 1) (6 t^{5} - 1)$ .

$5 ((6 t^{5} - 1) (6 t^{5} - 1)) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.5

Expand $(6 t^{5} - 1) (6 t^{5} - 1)$ using the FOIL Method.

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Step 10.5.1

Apply the distributive property.

$5 (6 t^{5} (6 t^{5} - 1) - 1 (6 t^{5} - 1)) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.5.2

Apply the distributive property.

$5 (6 t^{5} (6 t^{5}) + 6 t^{5} \cdot - 1 - 1 (6 t^{5} - 1)) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.5.3

Apply the distributive property.

$5 (6 t^{5} (6 t^{5}) + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6

Simplify and combine like terms.

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Step 10.6.1

Simplify each term.

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Step 10.6.1.1

Rewrite using the commutative property of multiplication.

$5 (6 \cdot 6 t^{5} t^{5} + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.2

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

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Step 10.6.1.2.1

Move $t^{5}$ .

$5 (6 \cdot 6 (t^{5} t^{5}) + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.2.2

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

$5 (6 \cdot 6 t^{5 + 5} + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.2.3

Add $5$ and $5$ .

$5 (6 \cdot 6 t^{10} + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.3

Multiply $6$ by $6$ .

$5 (36 t^{10} + 6 t^{5} \cdot - 1 - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.4

Multiply $- 1$ by $6$ .

$5 (36 t^{10} - 6 t^{5} - 1 (6 t^{5}) - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.5

Multiply $6$ by $- 1$ .

$5 (36 t^{10} - 6 t^{5} - 6 t^{5} - 1 \cdot - 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.1.6

Multiply $- 1$ by $- 1$ .

$5 (36 t^{10} - 6 t^{5} - 6 t^{5} + 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.6.2

Subtract $6 t^{5}$ from $- 6 t^{5}$ .

$5 (36 t^{10} - 12 t^{5} + 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.7

Apply the distributive property.

$(5 (36 t^{10}) + 5 (- 12 t^{5}) + 5 \cdot 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.8

Simplify.

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Step 10.8.1

Multiply $36$ by $5$ .

$(180 t^{10} + 5 (- 12 t^{5}) + 5 \cdot 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.8.2

Multiply $- 12$ by $5$ .

$(180 t^{10} - 60 t^{5} + 5 \cdot 1) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.8.3

Multiply $5$ by $1$ .

$(180 t^{10} - 60 t^{5} + 5) (- 90 t^{5} - (6 t^{5} - 1))$

Step 10.9

Simplify each term.

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Step 10.9.1

Apply the distributive property.

$(180 t^{10} - 60 t^{5} + 5) (- 90 t^{5} - (6 t^{5}) - - 1)$

Step 10.9.2

Multiply $6$ by $- 1$ .

$(180 t^{10} - 60 t^{5} + 5) (- 90 t^{5} - 6 t^{5} - - 1)$

Step 10.9.3

Multiply $- 1$ by $- 1$ .

$(180 t^{10} - 60 t^{5} + 5) (- 90 t^{5} - 6 t^{5} + 1)$

Step 10.10

Subtract $6 t^{5}$ from $- 90 t^{5}$ .

$(180 t^{10} - 60 t^{5} + 5) (- 96 t^{5} + 1)$

Step 10.11

Expand $(180 t^{10} - 60 t^{5} + 5) (- 96 t^{5} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$180 t^{10} (- 96 t^{5}) + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12

Simplify each term.

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Step 10.12.1

Rewrite using the commutative property of multiplication.

$180 \cdot - 96 t^{10} t^{5} + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.2

Multiply $t^{10}$ by $t^{5}$ by adding the exponents.

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Step 10.12.2.1

Move $t^{5}$ .

$180 \cdot - 96 (t^{5} t^{10}) + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.2.2

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

$180 \cdot - 96 t^{5 + 10} + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.2.3

Add $5$ and $10$ .

$180 \cdot - 96 t^{15} + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.3

Multiply $180$ by $- 96$ .

$- 17280 t^{15} + 180 t^{10} \cdot 1 - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.4

Multiply $180$ by $1$ .

$- 17280 t^{15} + 180 t^{10} - 60 t^{5} (- 96 t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.5

Rewrite using the commutative property of multiplication.

$- 17280 t^{15} + 180 t^{10} - 60 \cdot - 96 t^{5} t^{5} - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.6

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

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Step 10.12.6.1

Move $t^{5}$ .

$- 17280 t^{15} + 180 t^{10} - 60 \cdot - 96 (t^{5} t^{5}) - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.6.2

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

$- 17280 t^{15} + 180 t^{10} - 60 \cdot - 96 t^{5 + 5} - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.6.3

Add $5$ and $5$ .

$- 17280 t^{15} + 180 t^{10} - 60 \cdot - 96 t^{10} - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.7

Multiply $- 60$ by $- 96$ .

$- 17280 t^{15} + 180 t^{10} + 5760 t^{10} - 60 t^{5} \cdot 1 + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.8

Multiply $- 60$ by $1$ .

$- 17280 t^{15} + 180 t^{10} + 5760 t^{10} - 60 t^{5} + 5 (- 96 t^{5}) + 5 \cdot 1$

Step 10.12.9

Multiply $- 96$ by $5$ .

$- 17280 t^{15} + 180 t^{10} + 5760 t^{10} - 60 t^{5} - 480 t^{5} + 5 \cdot 1$

Step 10.12.10

Multiply $5$ by $1$ .

$- 17280 t^{15} + 180 t^{10} + 5760 t^{10} - 60 t^{5} - 480 t^{5} + 5$

Step 10.13

Add $180 t^{10}$ and $5760 t^{10}$ .

$- 17280 t^{15} + 5940 t^{10} - 60 t^{5} - 480 t^{5} + 5$

Step 10.14

Subtract $480 t^{5}$ from $- 60 t^{5}$ .

$- 17280 t^{15} + 5940 t^{10} - 540 t^{5} + 5$