Calculus Examples

$5500 {(1 - \frac{t}{50})}^{2}$

Step 1

Rewrite ${(1 - \frac{t}{50})}^{2}$ as $(1 - \frac{t}{50}) (1 - \frac{t}{50})$ .

$\frac{d}{d t} [5500 ((1 - \frac{t}{50}) (1 - \frac{t}{50}))]$

Step 2

Expand $(1 - \frac{t}{50}) (1 - \frac{t}{50})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d t} [5500 (1 (1 - \frac{t}{50}) - \frac{t}{50} (1 - \frac{t}{50}))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d t} [5500 (1 \cdot 1 + 1 (- \frac{t}{50}) - \frac{t}{50} (1 - \frac{t}{50}))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d t} [5500 (1 \cdot 1 + 1 (- \frac{t}{50}) - \frac{t}{50} \cdot 1 - \frac{t}{50} (- \frac{t}{50}))]$

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 $1$ by $1$ .

$\frac{d}{d t} [5500 (1 + 1 (- \frac{t}{50}) - \frac{t}{50} \cdot 1 - \frac{t}{50} (- \frac{t}{50}))]$

Step 3.1.2

Multiply $- \frac{t}{50}$ by $1$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} \cdot 1 - \frac{t}{50} (- \frac{t}{50}))]$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} - \frac{t}{50} (- \frac{t}{50}))]$

Step 3.1.4

Multiply $- \frac{t}{50} (- \frac{t}{50})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + 1 \frac{t}{50} \frac{t}{50})]$

Step 3.1.4.2

Multiply $\frac{t}{50}$ by $1$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t}{50} \cdot \frac{t}{50})]$

Step 3.1.4.3

Multiply $\frac{t}{50}$ by $\frac{t}{50}$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t \cdot t}{50 \cdot 50})]$

Step 3.1.4.4

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

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t^{1} t}{50 \cdot 50})]$

Step 3.1.4.5

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

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t^{1} t^{1}}{50 \cdot 50})]$

Step 3.1.4.6

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

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t^{1 + 1}}{50 \cdot 50})]$

Step 3.1.4.7

Add $1$ and $1$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t^{2}}{50 \cdot 50})]$

Step 3.1.4.8

Multiply $50$ by $50$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{50} - \frac{t}{50} + \frac{t^{2}}{2500})]$

Step 3.2

Subtract $\frac{t}{50}$ from $- \frac{t}{50}$ .

$\frac{d}{d t} [5500 (1 - 2 \frac{t}{50} + \frac{t^{2}}{2500})]$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$\frac{d}{d t} [5500 (1 + 2 (- 1) \frac{t}{50} + \frac{t^{2}}{2500})]$

Step 4.1.2

Factor $2$ out of $50$ .

$\frac{d}{d t} [5500 (1 + 2 \cdot - 1 \frac{t}{2 \cdot 25} + \frac{t^{2}}{2500})]$

Step 4.1.3

Cancel the common factor.

$\frac{d}{d t} [5500 (1 + 2 \cdot - 1 \frac{t}{2 \cdot 25} + \frac{t^{2}}{2500})]$

Step 4.1.4

Rewrite the expression.

$\frac{d}{d t} [5500 (1 - 1 \frac{t}{25} + \frac{t^{2}}{2500})]$

Step 4.2

Rewrite $- 1 \frac{t}{25}$ as $- \frac{t}{25}$ .

$\frac{d}{d t} [5500 (1 - \frac{t}{25} + \frac{t^{2}}{2500})]$

Step 5

Since $5500$ is constant with respect to $t$ , the derivative of $5500 (1 - \frac{t}{25} + \frac{t^{2}}{2500})$ with respect to $t$ is $5500 \frac{d}{d t} [1 - \frac{t}{25} + \frac{t^{2}}{2500}]$ .

$5500 \frac{d}{d t} [1 - \frac{t}{25} + \frac{t^{2}}{2500}]$

Step 6

By the Sum Rule, the derivative of $1 - \frac{t}{25} + \frac{t^{2}}{2500}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- \frac{t}{25}] + \frac{d}{d t} [\frac{t^{2}}{2500}]$ .

$5500 (\frac{d}{d t} [1] + \frac{d}{d t} [- \frac{t}{25}] + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 7

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

$5500 (0 + \frac{d}{d t} [- \frac{t}{25}] + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 8

Add $0$ and $\frac{d}{d t} [- \frac{t}{25}]$ .

$5500 (\frac{d}{d t} [- \frac{t}{25}] + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 9

Since $- \frac{1}{25}$ is constant with respect to $t$ , the derivative of $- \frac{t}{25}$ with respect to $t$ is $- \frac{1}{25} \frac{d}{d t} [t]$ .

$5500 (- \frac{1}{25} \frac{d}{d t} [t] + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 10

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

$5500 (- \frac{1}{25} \cdot 1 + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 11

Multiply $- 1$ by $1$ .

$5500 (- \frac{1}{25} + \frac{d}{d t} [\frac{t^{2}}{2500}])$

Step 12

Since $\frac{1}{2500}$ is constant with respect to $t$ , the derivative of $\frac{t^{2}}{2500}$ with respect to $t$ is $\frac{1}{2500} \frac{d}{d t} [t^{2}]$ .

$5500 (- \frac{1}{25} + \frac{1}{2500} \frac{d}{d t} [t^{2}])$

Step 13

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

$5500 (- \frac{1}{25} + \frac{1}{2500} (2 t))$

Step 14

Simplify terms.

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Step 14.1

Combine $2$ and $\frac{1}{2500}$ .

$5500 (- \frac{1}{25} + \frac{2}{2500} t)$

Step 14.2

Combine $\frac{2}{2500}$ and $t$ .

$5500 (- \frac{1}{25} + \frac{2 t}{2500})$

Step 14.3

Cancel the common factor of $2$ and $2500$ .

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Step 14.3.1

Factor $2$ out of $2 t$ .

$5500 (- \frac{1}{25} + \frac{2 (t)}{2500})$

Step 14.3.2

Cancel the common factors.

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Step 14.3.2.1

Factor $2$ out of $2500$ .

$5500 (- \frac{1}{25} + \frac{2 t}{2 \cdot 1250})$

Step 14.3.2.2

Cancel the common factor.

$5500 (- \frac{1}{25} + \frac{2 t}{2 \cdot 1250})$

Step 14.3.2.3

Rewrite the expression.

$5500 (- \frac{1}{25} + \frac{t}{1250})$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$5500 (- \frac{1}{25}) + 5500 \frac{t}{1250}$

Step 15.2

Combine terms.

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Step 15.2.1

Multiply $- 1$ by $5500$ .

$- 5500 (\frac{1}{25}) + 5500 \frac{t}{1250}$

Step 15.2.2

Combine $- 5500$ and $\frac{1}{25}$ .

$\frac{- 5500}{25} + 5500 \frac{t}{1250}$

Step 15.2.3

Cancel the common factor of $- 5500$ and $25$ .

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Step 15.2.3.1

Factor $25$ out of $- 5500$ .

$\frac{25 \cdot - 220}{25} + 5500 \frac{t}{1250}$

Step 15.2.3.2

Cancel the common factors.

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Step 15.2.3.2.1

Factor $25$ out of $25$ .

$\frac{25 \cdot - 220}{25 (1)} + 5500 \frac{t}{1250}$

Step 15.2.3.2.2

Cancel the common factor.

$\frac{25 \cdot - 220}{25 \cdot 1} + 5500 \frac{t}{1250}$

Step 15.2.3.2.3

Rewrite the expression.

$\frac{- 220}{1} + 5500 \frac{t}{1250}$

Step 15.2.3.2.4

Divide $- 220$ by $1$ .

$- 220 + 5500 \frac{t}{1250}$

Step 15.2.4

Combine $5500$ and $\frac{t}{1250}$ .

$- 220 + \frac{5500 t}{1250}$

Step 15.2.5

Cancel the common factor of $5500$ and $1250$ .

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Step 15.2.5.1

Factor $250$ out of $5500 t$ .

$- 220 + \frac{250 (22 t)}{1250}$

Step 15.2.5.2

Cancel the common factors.

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Step 15.2.5.2.1

Factor $250$ out of $1250$ .

$- 220 + \frac{250 (22 t)}{250 (5)}$

Step 15.2.5.2.2

Cancel the common factor.

$- 220 + \frac{250 (22 t)}{250 \cdot 5}$

Step 15.2.5.2.3

Rewrite the expression.

$- 220 + \frac{22 t}{5}$

Step 15.3

Reorder terms.

$\frac{22 t}{5} - 220$