Calculus Examples

$y = t^{7} {(t^{5} - 6)}^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (t^{7} {(t^{5} - 6)}^{3})$

Step 2

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (t) = t^{7}$ and $g (t) = {(t^{5} - 6)}^{3}$ .

$t^{7} \frac{d}{d t} [{(t^{5} - 6)}^{3}] + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.2

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

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Step 3.2.1

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

$t^{7} (\frac{d}{d u} [u^{3}] \frac{d}{d t} [t^{5} - 6]) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.2.2

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

$t^{7} (3 u^{2} \frac{d}{d t} [t^{5} - 6]) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.2.3

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

$t^{7} (3 {(t^{5} - 6)}^{2} \frac{d}{d t} [t^{5} - 6]) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.3

Differentiate.

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Step 3.3.1

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

$t^{7} (3 {(t^{5} - 6)}^{2} (\frac{d}{d t} [t^{5}] + \frac{d}{d t} [- 6])) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.3.2

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

$t^{7} (3 {(t^{5} - 6)}^{2} (5 t^{4} + \frac{d}{d t} [- 6])) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.3.3

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6$ with respect to $t$ is $0$ .

$t^{7} (3 {(t^{5} - 6)}^{2} (5 t^{4} + 0)) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $5 t^{4}$ and $0$ .

$t^{7} (3 {(t^{5} - 6)}^{2} (5 t^{4})) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.3.4.2

Multiply $5$ by $3$ .

$t^{7} (15 {(t^{5} - 6)}^{2} t^{4}) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.4

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

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Step 3.4.1

Move $t^{4}$ .

$t^{4} t^{7} (15 {(t^{5} - 6)}^{2}) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.4.2

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

$t^{4 + 7} (15 {(t^{5} - 6)}^{2}) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.4.3

Add $4$ and $7$ .

$t^{11} (15 {(t^{5} - 6)}^{2}) + {(t^{5} - 6)}^{3} \frac{d}{d t} [t^{7}]$

Step 3.5

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

$t^{11} (15 {(t^{5} - 6)}^{2}) + {(t^{5} - 6)}^{3} (7 t^{6})$

Step 3.6

Move $7$ to the left of ${(t^{5} - 6)}^{3}$ .

$t^{11} (15 {(t^{5} - 6)}^{2}) + 7 \cdot {(t^{5} - 6)}^{3} t^{6}$

Step 3.7

Simplify.

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Step 3.7.1

Factor $t^{6} {(t^{5} - 6)}^{2}$ out of $t^{11} (15 {(t^{5} - 6)}^{2}) + 7 {(t^{5} - 6)}^{3} t^{6}$ .

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Step 3.7.1.1

Factor $t^{6} {(t^{5} - 6)}^{2}$ out of $t^{11} (15 {(t^{5} - 6)}^{2})$ .

$t^{6} {(t^{5} - 6)}^{2} (t^{5} \cdot (15)) + 7 {(t^{5} - 6)}^{3} t^{6}$

Step 3.7.1.2

Factor $t^{6} {(t^{5} - 6)}^{2}$ out of $7 {(t^{5} - 6)}^{3} t^{6}$ .

$t^{6} {(t^{5} - 6)}^{2} (t^{5} \cdot (15)) + t^{6} {(t^{5} - 6)}^{2} (7 (t^{5} - 6))$

Step 3.7.1.3

Factor $t^{6} {(t^{5} - 6)}^{2}$ out of $t^{6} {(t^{5} - 6)}^{2} (t^{5} \cdot (15)) + t^{6} {(t^{5} - 6)}^{2} (7 (t^{5} - 6))$ .

$t^{6} {(t^{5} - 6)}^{2} (t^{5} \cdot (15) + 7 (t^{5} - 6))$

Step 3.7.2

Move $15$ to the left of $t^{5}$ .

$t^{6} {(t^{5} - 6)}^{2} (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.3

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

$t^{6} ((t^{5} - 6) (t^{5} - 6)) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.4

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

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Step 3.7.4.1

Apply the distributive property.

$t^{6} (t^{5} (t^{5} - 6) - 6 (t^{5} - 6)) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.4.2

Apply the distributive property.

$t^{6} (t^{5} t^{5} + t^{5} \cdot - 6 - 6 (t^{5} - 6)) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.4.3

Apply the distributive property.

$t^{6} (t^{5} t^{5} + t^{5} \cdot - 6 - 6 t^{5} - 6 \cdot - 6) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.5

Simplify and combine like terms.

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Step 3.7.5.1

Simplify each term.

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Step 3.7.5.1.1

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

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Step 3.7.5.1.1.1

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

$t^{6} (t^{5 + 5} + t^{5} \cdot - 6 - 6 t^{5} - 6 \cdot - 6) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.5.1.1.2

Add $5$ and $5$ .

$t^{6} (t^{10} + t^{5} \cdot - 6 - 6 t^{5} - 6 \cdot - 6) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.5.1.2

Move $- 6$ to the left of $t^{5}$ .

$t^{6} (t^{10} - 6 \cdot t^{5} - 6 t^{5} - 6 \cdot - 6) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.5.1.3

Multiply $- 6$ by $- 6$ .

$t^{6} (t^{10} - 6 t^{5} - 6 t^{5} + 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.5.2

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

$t^{6} (t^{10} - 12 t^{5} + 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.6

Apply the distributive property.

$(t^{6} t^{10} + t^{6} (- 12 t^{5}) + t^{6} \cdot 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.7

Simplify.

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Step 3.7.7.1

Multiply $t^{6}$ by $t^{10}$ by adding the exponents.

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Step 3.7.7.1.1

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

$(t^{6 + 10} + t^{6} (- 12 t^{5}) + t^{6} \cdot 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.7.1.2

Add $6$ and $10$ .

$(t^{16} + t^{6} (- 12 t^{5}) + t^{6} \cdot 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.7.2

Rewrite using the commutative property of multiplication.

$(t^{16} - 12 t^{6} t^{5} + t^{6} \cdot 36) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.7.3

Move $36$ to the left of $t^{6}$ .

$(t^{16} - 12 t^{6} t^{5} + 36 \cdot t^{6}) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.8

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

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Step 3.7.8.1

Move $t^{5}$ .

$(t^{16} - 12 (t^{5} t^{6}) + 36 \cdot t^{6}) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.8.2

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

$(t^{16} - 12 t^{5 + 6} + 36 \cdot t^{6}) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.8.3

Add $5$ and $6$ .

$(t^{16} - 12 t^{11} + 36 \cdot t^{6}) (15 t^{5} + 7 (t^{5} - 6))$

$(t^{16} - 12 t^{11} + 36 t^{6}) (15 t^{5} + 7 (t^{5} - 6))$

Step 3.7.9

Simplify each term.

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Step 3.7.9.1

Apply the distributive property.

$(t^{16} - 12 t^{11} + 36 t^{6}) (15 t^{5} + 7 t^{5} + 7 \cdot - 6)$

Step 3.7.9.2

Multiply $7$ by $- 6$ .

$(t^{16} - 12 t^{11} + 36 t^{6}) (15 t^{5} + 7 t^{5} - 42)$

Step 3.7.10

Add $15 t^{5}$ and $7 t^{5}$ .

$(t^{16} - 12 t^{11} + 36 t^{6}) (22 t^{5} - 42)$

Step 3.7.11

Expand $(t^{16} - 12 t^{11} + 36 t^{6}) (22 t^{5} - 42)$ by multiplying each term in the first expression by each term in the second expression.

$t^{16} (22 t^{5}) + t^{16} \cdot - 42 - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12

Simplify each term.

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Step 3.7.12.1

Rewrite using the commutative property of multiplication.

$22 t^{16} t^{5} + t^{16} \cdot - 42 - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.2

Multiply $t^{16}$ by $t^{5}$ by adding the exponents.

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Step 3.7.12.2.1

Move $t^{5}$ .

$22 (t^{5} t^{16}) + t^{16} \cdot - 42 - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.2.2

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

$22 t^{5 + 16} + t^{16} \cdot - 42 - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.2.3

Add $5$ and $16$ .

$22 t^{21} + t^{16} \cdot - 42 - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.3

Move $- 42$ to the left of $t^{16}$ .

$22 t^{21} - 42 \cdot t^{16} - 12 t^{11} (22 t^{5}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.4

Rewrite using the commutative property of multiplication.

$22 t^{21} - 42 t^{16} - 12 \cdot 22 t^{11} t^{5} - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.5

Multiply $t^{11}$ by $t^{5}$ by adding the exponents.

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Step 3.7.12.5.1

Move $t^{5}$ .

$22 t^{21} - 42 t^{16} - 12 \cdot 22 (t^{5} t^{11}) - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.5.2

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

$22 t^{21} - 42 t^{16} - 12 \cdot 22 t^{5 + 11} - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.5.3

Add $5$ and $11$ .

$22 t^{21} - 42 t^{16} - 12 \cdot 22 t^{16} - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.6

Multiply $- 12$ by $22$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} - 12 t^{11} \cdot - 42 + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.7

Multiply $- 42$ by $- 12$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 36 t^{6} (22 t^{5}) + 36 t^{6} \cdot - 42$

Step 3.7.12.8

Rewrite using the commutative property of multiplication.

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 36 \cdot 22 t^{6} t^{5} + 36 t^{6} \cdot - 42$

Step 3.7.12.9

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

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Step 3.7.12.9.1

Move $t^{5}$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 36 \cdot 22 (t^{5} t^{6}) + 36 t^{6} \cdot - 42$

Step 3.7.12.9.2

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

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 36 \cdot 22 t^{5 + 6} + 36 t^{6} \cdot - 42$

Step 3.7.12.9.3

Add $5$ and $6$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 36 \cdot 22 t^{11} + 36 t^{6} \cdot - 42$

Step 3.7.12.10

Multiply $36$ by $22$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 792 t^{11} + 36 t^{6} \cdot - 42$

Step 3.7.12.11

Multiply $- 42$ by $36$ .

$22 t^{21} - 42 t^{16} - 264 t^{16} + 504 t^{11} + 792 t^{11} - 1512 t^{6}$

Step 3.7.13

Subtract $264 t^{16}$ from $- 42 t^{16}$ .

$22 t^{21} - 306 t^{16} + 504 t^{11} + 792 t^{11} - 1512 t^{6}$

Step 3.7.14

Add $504 t^{11}$ and $792 t^{11}$ .

$22 t^{21} - 306 t^{16} + 1296 t^{11} - 1512 t^{6}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 22 t^{21} - 306 t^{16} + 1296 t^{11} - 1512 t^{6}$

Step 5

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = 22 t^{21} - 306 t^{16} + 1296 t^{11} - 1512 t^{6}$