Calculus Examples

$g (t) = {(2 t^{2} - 1)}^{2} (3 t^{2})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move $3$ to the left of ${(2 t^{2} - 1)}^{2}$ .

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

Step 1.2

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

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

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 $f (t) = {(2 t^{2} - 1)}^{2}$ and $g (t) = t^{2}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

Move $2$ to the left of ${(2 t^{2} - 1)}^{2}$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = 2 t^{2} - 1$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $2 t^{2} - 1$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $2 t^{2} - 1$ .

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

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $2 t^{2} - 1$ with respect to $t$ is $\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [- 1]$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Multiply $2$ by $2$ .

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

Step 5.5

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

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{2} (2 (2 t^{2} - 1) (4 t + 0)))$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $4 t$ and $0$ .

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{2} (2 (2 t^{2} - 1) (4 t)))$

Step 5.6.2

Multiply $4$ by $2$ .

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{2} (8 (2 t^{2} - 1) t))$

Step 6

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

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Step 6.1

Move $t$ .

$3 (2 {(2 t^{2} - 1)}^{2} t + t \cdot t^{2} (8 (2 t^{2} - 1)))$

Step 6.2

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

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Step 6.2.1

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

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{1} t^{2} (8 (2 t^{2} - 1)))$

Step 6.2.2

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

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{1 + 2} (8 (2 t^{2} - 1)))$

Step 6.3

Add $1$ and $2$ .

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{3} (8 (2 t^{2} - 1)))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{3} (8 (2 t^{2}) + 8 \cdot - 1))$

Step 7.2

Apply the distributive property.

$3 (2 {(2 t^{2} - 1)}^{2} t + t^{3} (8 (2 t^{2})) + t^{3} (8 \cdot - 1))$

Step 7.3

Apply the distributive property.

$3 (2 {(2 t^{2} - 1)}^{2} t) + 3 (t^{3} (8 (2 t^{2}))) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4

Combine terms.

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Step 7.4.1

Multiply $2$ by $3$ .

$6 ({(2 t^{2} - 1)}^{2} t) + 3 (t^{3} (8 (2 t^{2}))) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.2

Multiply $2$ by $8$ .

$6 {(2 t^{2} - 1)}^{2} t + 3 (t^{3} (16 t^{2})) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.3

Multiply $t^{3}$ by $t^{2}$ by adding the exponents.

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Step 7.4.3.1

Move $t^{2}$ .

$6 {(2 t^{2} - 1)}^{2} t + 3 (t^{2} t^{3} \cdot 16) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.3.2

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

$6 {(2 t^{2} - 1)}^{2} t + 3 (t^{2 + 3} \cdot 16) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.3.3

Add $2$ and $3$ .

$6 {(2 t^{2} - 1)}^{2} t + 3 (t^{5} \cdot 16) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.4

Move $16$ to the left of $t^{5}$ .

$6 {(2 t^{2} - 1)}^{2} t + 3 (16 \cdot t^{5}) + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.5

Multiply $16$ by $3$ .

$6 {(2 t^{2} - 1)}^{2} t + 48 t^{5} + 3 (t^{3} (8 \cdot - 1))$

Step 7.4.6

Multiply $8$ by $- 1$ .

$6 {(2 t^{2} - 1)}^{2} t + 48 t^{5} + 3 (t^{3} \cdot - 8)$

Step 7.4.7

Move $- 8$ to the left of $t^{3}$ .

$6 {(2 t^{2} - 1)}^{2} t + 48 t^{5} + 3 (- 8 \cdot t^{3})$

Step 7.4.8

Multiply $- 8$ by $3$ .

$6 {(2 t^{2} - 1)}^{2} t + 48 t^{5} - 24 t^{3}$

Step 7.5

Reorder terms.

$48 t^{5} - 24 t^{3} + 6 t {(2 t^{2} - 1)}^{2}$

Step 7.6

Simplify each term.

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Step 7.6.1

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

$48 t^{5} - 24 t^{3} + 6 t ((2 t^{2} - 1) (2 t^{2} - 1))$

Step 7.6.2

Expand $(2 t^{2} - 1) (2 t^{2} - 1)$ using the FOIL Method.

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Step 7.6.2.1

Apply the distributive property.

$48 t^{5} - 24 t^{3} + 6 t (2 t^{2} (2 t^{2} - 1) - 1 (2 t^{2} - 1))$

Step 7.6.2.2

Apply the distributive property.

$48 t^{5} - 24 t^{3} + 6 t (2 t^{2} (2 t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2} - 1))$

Step 7.6.2.3

Apply the distributive property.

$48 t^{5} - 24 t^{3} + 6 t (2 t^{2} (2 t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3

Simplify and combine like terms.

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Step 7.6.3.1

Simplify each term.

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Step 7.6.3.1.1

Rewrite using the commutative property of multiplication.

$48 t^{5} - 24 t^{3} + 6 t (2 \cdot 2 t^{2} t^{2} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.2

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

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Step 7.6.3.1.2.1

Move $t^{2}$ .

$48 t^{5} - 24 t^{3} + 6 t (2 \cdot 2 (t^{2} t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.2.2

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

$48 t^{5} - 24 t^{3} + 6 t (2 \cdot 2 t^{2 + 2} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.2.3

Add $2$ and $2$ .

$48 t^{5} - 24 t^{3} + 6 t (2 \cdot 2 t^{4} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.3

Multiply $2$ by $2$ .

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.4

Multiply $- 1$ by $2$ .

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4} - 2 t^{2} - 1 (2 t^{2}) - 1 \cdot - 1)$

Step 7.6.3.1.5

Multiply $2$ by $- 1$ .

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4} - 2 t^{2} - 2 t^{2} - 1 \cdot - 1)$

Step 7.6.3.1.6

Multiply $- 1$ by $- 1$ .

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4} - 2 t^{2} - 2 t^{2} + 1)$

Step 7.6.3.2

Subtract $2 t^{2}$ from $- 2 t^{2}$ .

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4} - 4 t^{2} + 1)$

Step 7.6.4

Apply the distributive property.

$48 t^{5} - 24 t^{3} + 6 t (4 t^{4}) + 6 t (- 4 t^{2}) + 6 t \cdot 1$

Step 7.6.5

Simplify.

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Step 7.6.5.1

Rewrite using the commutative property of multiplication.

$48 t^{5} - 24 t^{3} + 6 \cdot 4 t \cdot t^{4} + 6 t (- 4 t^{2}) + 6 t \cdot 1$

Step 7.6.5.2

Rewrite using the commutative property of multiplication.

$48 t^{5} - 24 t^{3} + 6 \cdot 4 t \cdot t^{4} + 6 \cdot - 4 t \cdot t^{2} + 6 t \cdot 1$

Step 7.6.5.3

Multiply $6$ by $1$ .

$48 t^{5} - 24 t^{3} + 6 \cdot 4 t \cdot t^{4} + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6

Simplify each term.

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Step 7.6.6.1

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

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Step 7.6.6.1.1

Move $t^{4}$ .

$48 t^{5} - 24 t^{3} + 6 \cdot 4 (t^{4} t) + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6.1.2

Multiply $t^{4}$ by $t$ .

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Step 7.6.6.1.2.1

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

$48 t^{5} - 24 t^{3} + 6 \cdot 4 (t^{4} t^{1}) + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6.1.2.2

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

$48 t^{5} - 24 t^{3} + 6 \cdot 4 t^{4 + 1} + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6.1.3

Add $4$ and $1$ .

$48 t^{5} - 24 t^{3} + 6 \cdot 4 t^{5} + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6.2

Multiply $6$ by $4$ .

$48 t^{5} - 24 t^{3} + 24 t^{5} + 6 \cdot - 4 t \cdot t^{2} + 6 t$

Step 7.6.6.3

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

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Step 7.6.6.3.1

Move $t^{2}$ .

$48 t^{5} - 24 t^{3} + 24 t^{5} + 6 \cdot - 4 (t^{2} t) + 6 t$

Step 7.6.6.3.2

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

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Step 7.6.6.3.2.1

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

$48 t^{5} - 24 t^{3} + 24 t^{5} + 6 \cdot - 4 (t^{2} t^{1}) + 6 t$

Step 7.6.6.3.2.2

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

$48 t^{5} - 24 t^{3} + 24 t^{5} + 6 \cdot - 4 t^{2 + 1} + 6 t$

Step 7.6.6.3.3

Add $2$ and $1$ .

$48 t^{5} - 24 t^{3} + 24 t^{5} + 6 \cdot - 4 t^{3} + 6 t$

Step 7.6.6.4

Multiply $6$ by $- 4$ .

$48 t^{5} - 24 t^{3} + 24 t^{5} - 24 t^{3} + 6 t$

Step 7.7

Add $48 t^{5}$ and $24 t^{5}$ .

$72 t^{5} - 24 t^{3} - 24 t^{3} + 6 t$

Step 7.8

Subtract $24 t^{3}$ from $- 24 t^{3}$ .

$72 t^{5} - 48 t^{3} + 6 t$