Calculus Examples

$f (x) = 30 {(65 + 5 t)}^{2} - 1300 t$

Step 1

Rewrite ${(65 + 5 t)}^{2}$ as $(65 + 5 t) (65 + 5 t)$ .

$\frac{d}{d t} [30 ((65 + 5 t) (65 + 5 t)) - 1300 t]$

Step 2

Expand $(65 + 5 t) (65 + 5 t)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d t} [30 (65 (65 + 5 t) + 5 t (65 + 5 t)) - 1300 t]$

Step 2.2

Apply the distributive property.

$\frac{d}{d t} [30 (65 \cdot 65 + 65 (5 t) + 5 t (65 + 5 t)) - 1300 t]$

Step 2.3

Apply the distributive property.

$\frac{d}{d t} [30 (65 \cdot 65 + 65 (5 t) + 5 t \cdot 65 + 5 t (5 t)) - 1300 t]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $65$ by $65$ .

$\frac{d}{d t} [30 (4225 + 65 (5 t) + 5 t \cdot 65 + 5 t (5 t)) - 1300 t]$

Step 3.1.2

Multiply $5$ by $65$ .

$\frac{d}{d t} [30 (4225 + 325 t + 5 t \cdot 65 + 5 t (5 t)) - 1300 t]$

Step 3.1.3

Multiply $65$ by $5$ .

$\frac{d}{d t} [30 (4225 + 325 t + 325 t + 5 t (5 t)) - 1300 t]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d t} [30 (4225 + 325 t + 325 t + 5 \cdot 5 t \cdot t) - 1300 t]$

Step 3.1.5

Multiply $t$ by $t$ by adding the exponents.

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Step 3.1.5.1

Move $t$ .

$\frac{d}{d t} [30 (4225 + 325 t + 325 t + 5 \cdot 5 (t \cdot t)) - 1300 t]$

Step 3.1.5.2

Multiply $t$ by $t$ .

$\frac{d}{d t} [30 (4225 + 325 t + 325 t + 5 \cdot 5 t^{2}) - 1300 t]$

Step 3.1.6

Multiply $5$ by $5$ .

$\frac{d}{d t} [30 (4225 + 325 t + 325 t + 25 t^{2}) - 1300 t]$

Step 3.2

Add $325 t$ and $325 t$ .

$\frac{d}{d t} [30 (4225 + 650 t + 25 t^{2}) - 1300 t]$

Step 4

By the Sum Rule, the derivative of $30 (4225 + 650 t + 25 t^{2}) - 1300 t$ with respect to $t$ is $\frac{d}{d t} [30 (4225 + 650 t + 25 t^{2})] + \frac{d}{d t} [- 1300 t]$ .

$\frac{d}{d t} [30 (4225 + 650 t + 25 t^{2})] + \frac{d}{d t} [- 1300 t]$

Step 5

Evaluate $\frac{d}{d t} [30 (4225 + 650 t + 25 t^{2})]$ .

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Step 5.1

Since $30$ is constant with respect to $t$ , the derivative of $30 (4225 + 650 t + 25 t^{2})$ with respect to $t$ is $30 \frac{d}{d t} [4225 + 650 t + 25 t^{2}]$ .

$30 \frac{d}{d t} [4225 + 650 t + 25 t^{2}] + \frac{d}{d t} [- 1300 t]$

Step 5.2

By the Sum Rule, the derivative of $4225 + 650 t + 25 t^{2}$ with respect to $t$ is $\frac{d}{d t} [4225] + \frac{d}{d t} [650 t] + \frac{d}{d t} [25 t^{2}]$ .

$30 (\frac{d}{d t} [4225] + \frac{d}{d t} [650 t] + \frac{d}{d t} [25 t^{2}]) + \frac{d}{d t} [- 1300 t]$

Step 5.3

Since $4225$ is constant with respect to $t$ , the derivative of $4225$ with respect to $t$ is $0$ .

$30 (0 + \frac{d}{d t} [650 t] + \frac{d}{d t} [25 t^{2}]) + \frac{d}{d t} [- 1300 t]$

Step 5.4

Since $650$ is constant with respect to $t$ , the derivative of $650 t$ with respect to $t$ is $650 \frac{d}{d t} [t]$ .

$30 (0 + 650 \frac{d}{d t} [t] + \frac{d}{d t} [25 t^{2}]) + \frac{d}{d t} [- 1300 t]$

Step 5.5

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

$30 (0 + 650 \cdot 1 + \frac{d}{d t} [25 t^{2}]) + \frac{d}{d t} [- 1300 t]$

Step 5.6

Since $25$ is constant with respect to $t$ , the derivative of $25 t^{2}$ with respect to $t$ is $25 \frac{d}{d t} [t^{2}]$ .

$30 (0 + 650 \cdot 1 + 25 \frac{d}{d t} [t^{2}]) + \frac{d}{d t} [- 1300 t]$

Step 5.7

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

$30 (0 + 650 \cdot 1 + 25 (2 t)) + \frac{d}{d t} [- 1300 t]$

Step 5.8

Multiply $650$ by $1$ .

$30 (0 + 650 + 25 (2 t)) + \frac{d}{d t} [- 1300 t]$

Step 5.9

Add $0$ and $650$ .

$30 (650 + 25 (2 t)) + \frac{d}{d t} [- 1300 t]$

Step 5.10

Multiply $2$ by $25$ .

$30 (650 + 50 t) + \frac{d}{d t} [- 1300 t]$

Step 6

Evaluate $\frac{d}{d t} [- 1300 t]$ .

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Step 6.1

Since $- 1300$ is constant with respect to $t$ , the derivative of $- 1300 t$ with respect to $t$ is $- 1300 \frac{d}{d t} [t]$ .

$30 (650 + 50 t) - 1300 \frac{d}{d t} [t]$

Step 6.2

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

$30 (650 + 50 t) - 1300 \cdot 1$

Step 6.3

Multiply $- 1300$ by $1$ .

$30 (650 + 50 t) - 1300$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$30 \cdot 650 + 30 (50 t) - 1300$

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $30$ by $650$ .

$19500 + 30 (50 t) - 1300$

Step 7.2.2

Multiply $50$ by $30$ .

$19500 + 1500 t - 1300$

Step 7.2.3

Subtract $1300$ from $19500$ .

$1500 t + 18200$