Calculus Examples

$(5 t^{3} + t^{2}) (2 t - 4)$

Step 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 $f (t) = 5 t^{3} + t^{2}$ and $g (t) = 2 t - 4$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 t - 4$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [- 4]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 2.5

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4$ with respect to $t$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2$ and $0$ .

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

Step 2.6.2

Move $2$ to the left of $5 t^{3} + t^{2}$ .

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

Step 2.7

By the Sum Rule, the derivative of $5 t^{3} + t^{2}$ with respect to $t$ is $\frac{d}{d t} [5 t^{3}] + \frac{d}{d t} [t^{2}]$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $3$ by $5$ .

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

Step 2.11

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

$2 (5 t^{3} + t^{2}) + (2 t - 4) (15 t^{2} + 2 t)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$2 (5 t^{3}) + 2 t^{2} + (2 t - 4) (15 t^{2} + 2 t)$

Step 3.2

Apply the distributive property.

$2 (5 t^{3}) + 2 t^{2} + (2 t - 4) (15 t^{2}) + (2 t - 4) (2 t)$

Step 3.3

Apply the distributive property.

$2 (5 t^{3}) + 2 t^{2} + 2 t (15 t^{2}) - 4 (15 t^{2}) + (2 t - 4) (2 t)$

Step 3.4

Apply the distributive property.

$2 (5 t^{3}) + 2 t^{2} + 2 t (15 t^{2}) - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $5$ by $2$ .

$10 t^{3} + 2 t^{2} + 2 t (15 t^{2}) - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.2

Multiply $15$ by $2$ .

$10 t^{3} + 2 t^{2} + 30 t \cdot t^{2} - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.3

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

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Step 3.5.3.1

Move $t^{2}$ .

$10 t^{3} + 2 t^{2} + 30 (t^{2} t) - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.3.2

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

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Step 3.5.3.2.1

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

$10 t^{3} + 2 t^{2} + 30 (t^{2} t^{1}) - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.3.2.2

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

$10 t^{3} + 2 t^{2} + 30 t^{2 + 1} - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.3.3

Add $2$ and $1$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 4 (15 t^{2}) + 2 t (2 t) - 4 (2 t)$

Step 3.5.4

Multiply $15$ by $- 4$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 2 t (2 t) - 4 (2 t)$

Step 3.5.5

Multiply $2$ by $2$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 t \cdot t - 4 (2 t)$

Step 3.5.6

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

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 (t^{1} t) - 4 (2 t)$

Step 3.5.7

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

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 (t^{1} t^{1}) - 4 (2 t)$

Step 3.5.8

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

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 t^{1 + 1} - 4 (2 t)$

Step 3.5.9

Add $1$ and $1$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 t^{2} - 4 (2 t)$

Step 3.5.10

Multiply $2$ by $- 4$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 60 t^{2} + 4 t^{2} - 8 t$

Step 3.5.11

Add $- 60 t^{2}$ and $4 t^{2}$ .

$10 t^{3} + 2 t^{2} + 30 t^{3} - 56 t^{2} - 8 t$

Step 3.5.12

Add $10 t^{3}$ and $30 t^{3}$ .

$40 t^{3} + 2 t^{2} - 56 t^{2} - 8 t$

Step 3.5.13

Subtract $56 t^{2}$ from $2 t^{2}$ .

$40 t^{3} - 54 t^{2} - 8 t$