Calculus Examples

$R (x) = 2000 {(1 - \frac{x}{600})}^{2}$

Step 1

Rewrite ${(1 - \frac{x}{600})}^{2}$ as $(1 - \frac{x}{600}) (1 - \frac{x}{600})$ .

$\frac{d}{d x} [2000 ((1 - \frac{x}{600}) (1 - \frac{x}{600}))]$

Step 2

Expand $(1 - \frac{x}{600}) (1 - \frac{x}{600})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [2000 (1 (1 - \frac{x}{600}) - \frac{x}{600} (1 - \frac{x}{600}))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [2000 (1 \cdot 1 + 1 (- \frac{x}{600}) - \frac{x}{600} (1 - \frac{x}{600}))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [2000 (1 \cdot 1 + 1 (- \frac{x}{600}) - \frac{x}{600} \cdot 1 - \frac{x}{600} (- \frac{x}{600}))]$

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 $1$ .

$\frac{d}{d x} [2000 (1 + 1 (- \frac{x}{600}) - \frac{x}{600} \cdot 1 - \frac{x}{600} (- \frac{x}{600}))]$

Step 3.1.2

Multiply $- \frac{x}{600}$ by $1$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} \cdot 1 - \frac{x}{600} (- \frac{x}{600}))]$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} - \frac{x}{600} (- \frac{x}{600}))]$

Step 3.1.4

Multiply $- \frac{x}{600} (- \frac{x}{600})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + 1 \frac{x}{600} \frac{x}{600})]$

Step 3.1.4.2

Multiply $\frac{x}{600}$ by $1$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x}{600} \cdot \frac{x}{600})]$

Step 3.1.4.3

Multiply $\frac{x}{600}$ by $\frac{x}{600}$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x \cdot x}{600 \cdot 600})]$

Step 3.1.4.4

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

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x^{1} x}{600 \cdot 600})]$

Step 3.1.4.5

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

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x^{1} x^{1}}{600 \cdot 600})]$

Step 3.1.4.6

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

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x^{1 + 1}}{600 \cdot 600})]$

Step 3.1.4.7

Add $1$ and $1$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x^{2}}{600 \cdot 600})]$

Step 3.1.4.8

Multiply $600$ by $600$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{600} - \frac{x}{600} + \frac{x^{2}}{360000})]$

Step 3.2

Subtract $\frac{x}{600}$ from $- \frac{x}{600}$ .

$\frac{d}{d x} [2000 (1 - 2 \frac{x}{600} + \frac{x^{2}}{360000})]$

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 $- 2$ .

$\frac{d}{d x} [2000 (1 + 2 (- 1) \frac{x}{600} + \frac{x^{2}}{360000})]$

Step 4.1.2

Factor $2$ out of $600$ .

$\frac{d}{d x} [2000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 300} + \frac{x^{2}}{360000})]$

Step 4.1.3

Cancel the common factor.

$\frac{d}{d x} [2000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 300} + \frac{x^{2}}{360000})]$

Step 4.1.4

Rewrite the expression.

$\frac{d}{d x} [2000 (1 - 1 \frac{x}{300} + \frac{x^{2}}{360000})]$

Step 4.2

Rewrite $- 1 \frac{x}{300}$ as $- \frac{x}{300}$ .

$\frac{d}{d x} [2000 (1 - \frac{x}{300} + \frac{x^{2}}{360000})]$

Step 5

Since $2000$ is constant with respect to $x$ , the derivative of $2000 (1 - \frac{x}{300} + \frac{x^{2}}{360000})$ with respect to $x$ is $2000 \frac{d}{d x} [1 - \frac{x}{300} + \frac{x^{2}}{360000}]$ .

$2000 \frac{d}{d x} [1 - \frac{x}{300} + \frac{x^{2}}{360000}]$

Step 6

By the Sum Rule, the derivative of $1 - \frac{x}{300} + \frac{x^{2}}{360000}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{300}] + \frac{d}{d x} [\frac{x^{2}}{360000}]$ .

$2000 (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{300}] + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 7

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

$2000 (0 + \frac{d}{d x} [- \frac{x}{300}] + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 8

Add $0$ and $\frac{d}{d x} [- \frac{x}{300}]$ .

$2000 (\frac{d}{d x} [- \frac{x}{300}] + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 9

Since $- \frac{1}{300}$ is constant with respect to $x$ , the derivative of $- \frac{x}{300}$ with respect to $x$ is $- \frac{1}{300} \frac{d}{d x} [x]$ .

$2000 (- \frac{1}{300} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 10

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

$2000 (- \frac{1}{300} \cdot 1 + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 11

Multiply $- 1$ by $1$ .

$2000 (- \frac{1}{300} + \frac{d}{d x} [\frac{x^{2}}{360000}])$

Step 12

Since $\frac{1}{360000}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{360000}$ with respect to $x$ is $\frac{1}{360000} \frac{d}{d x} [x^{2}]$ .

$2000 (- \frac{1}{300} + \frac{1}{360000} \frac{d}{d x} [x^{2}])$

Step 13

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

$2000 (- \frac{1}{300} + \frac{1}{360000} (2 x))$

Step 14

Simplify terms.

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Step 14.1

Combine $2$ and $\frac{1}{360000}$ .

$2000 (- \frac{1}{300} + \frac{2}{360000} x)$

Step 14.2

Combine $\frac{2}{360000}$ and $x$ .

$2000 (- \frac{1}{300} + \frac{2 x}{360000})$

Step 14.3

Cancel the common factor of $2$ and $360000$ .

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Step 14.3.1

Factor $2$ out of $2 x$ .

$2000 (- \frac{1}{300} + \frac{2 (x)}{360000})$

Step 14.3.2

Cancel the common factors.

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Step 14.3.2.1

Factor $2$ out of $360000$ .

$2000 (- \frac{1}{300} + \frac{2 x}{2 \cdot 180000})$

Step 14.3.2.2

Cancel the common factor.

$2000 (- \frac{1}{300} + \frac{2 x}{2 \cdot 180000})$

Step 14.3.2.3

Rewrite the expression.

$2000 (- \frac{1}{300} + \frac{x}{180000})$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$2000 (- \frac{1}{300}) + 2000 \frac{x}{180000}$

Step 15.2

Combine terms.

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Step 15.2.1

Multiply $- 1$ by $2000$ .

$- 2000 (\frac{1}{300}) + 2000 \frac{x}{180000}$

Step 15.2.2

Combine $- 2000$ and $\frac{1}{300}$ .

$\frac{- 2000}{300} + 2000 \frac{x}{180000}$

Step 15.2.3

Cancel the common factor of $- 2000$ and $300$ .

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Step 15.2.3.1

Factor $100$ out of $- 2000$ .

$\frac{100 (- 20)}{300} + 2000 \frac{x}{180000}$

Step 15.2.3.2

Cancel the common factors.

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Step 15.2.3.2.1

Factor $100$ out of $300$ .

$\frac{100 \cdot - 20}{100 \cdot 3} + 2000 \frac{x}{180000}$

Step 15.2.3.2.2

Cancel the common factor.

$\frac{100 \cdot - 20}{100 \cdot 3} + 2000 \frac{x}{180000}$

Step 15.2.3.2.3

Rewrite the expression.

$\frac{- 20}{3} + 2000 \frac{x}{180000}$

Step 15.2.4

Move the negative in front of the fraction.

$- \frac{20}{3} + 2000 \frac{x}{180000}$

Step 15.2.5

Combine $2000$ and $\frac{x}{180000}$ .

$- \frac{20}{3} + \frac{2000 x}{180000}$

Step 15.2.6

Cancel the common factor of $2000$ and $180000$ .

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Step 15.2.6.1

Factor $2000$ out of $2000 x$ .

$- \frac{20}{3} + \frac{2000 (x)}{180000}$

Step 15.2.6.2

Cancel the common factors.

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Step 15.2.6.2.1

Factor $2000$ out of $180000$ .

$- \frac{20}{3} + \frac{2000 x}{2000 \cdot 90}$

Step 15.2.6.2.2

Cancel the common factor.

$- \frac{20}{3} + \frac{2000 x}{2000 \cdot 90}$

Step 15.2.6.2.3

Rewrite the expression.

$- \frac{20}{3} + \frac{x}{90}$

Step 15.3

Reorder terms.

$\frac{x}{90} - \frac{20}{3}$