Calculus Examples

$y = 4000 {(1 - \frac{x}{800})}^{2}$

Step 1

Rewrite ${(1 - \frac{x}{800})}^{2}$ as $(1 - \frac{x}{800}) (1 - \frac{x}{800})$ .

$\frac{d}{d x} [4000 ((1 - \frac{x}{800}) (1 - \frac{x}{800}))]$

Step 2

Expand $(1 - \frac{x}{800}) (1 - \frac{x}{800})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [4000 (1 (1 - \frac{x}{800}) - \frac{x}{800} (1 - \frac{x}{800}))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{800}) - \frac{x}{800} (1 - \frac{x}{800}))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{800}) - \frac{x}{800} \cdot 1 - \frac{x}{800} (- \frac{x}{800}))]$

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 $1$ .

$\frac{d}{d x} [4000 (1 + 1 (- \frac{x}{800}) - \frac{x}{800} \cdot 1 - \frac{x}{800} (- \frac{x}{800}))]$

Step 3.1.2

Multiply $- \frac{x}{800}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} \cdot 1 - \frac{x}{800} (- \frac{x}{800}))]$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} - \frac{x}{800} (- \frac{x}{800}))]$

Step 3.1.4

Multiply $- \frac{x}{800} (- \frac{x}{800})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + 1 \frac{x}{800} \frac{x}{800})]$

Step 3.1.4.2

Multiply $\frac{x}{800}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x}{800} \cdot \frac{x}{800})]$

Step 3.1.4.3

Multiply $\frac{x}{800}$ by $\frac{x}{800}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x \cdot x}{800 \cdot 800})]$

Step 3.1.4.4

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

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x^{1} x}{800 \cdot 800})]$

Step 3.1.4.5

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

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x^{1} x^{1}}{800 \cdot 800})]$

Step 3.1.4.6

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

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x^{1 + 1}}{800 \cdot 800})]$

Step 3.1.4.7

Add $1$ and $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x^{2}}{800 \cdot 800})]$

Step 3.1.4.8

Multiply $800$ by $800$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{800} - \frac{x}{800} + \frac{x^{2}}{640000})]$

Step 3.2

Subtract $\frac{x}{800}$ from $- \frac{x}{800}$ .

$\frac{d}{d x} [4000 (1 - 2 \frac{x}{800} + \frac{x^{2}}{640000})]$

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 $- 2$ .

$\frac{d}{d x} [4000 (1 + 2 (- 1) \frac{x}{800} + \frac{x^{2}}{640000})]$

Step 4.1.2

Factor $2$ out of $800$ .

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 400} + \frac{x^{2}}{640000})]$

Step 4.1.3

Cancel the common factor.

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 400} + \frac{x^{2}}{640000})]$

Step 4.1.4

Rewrite the expression.

$\frac{d}{d x} [4000 (1 - 1 \frac{x}{400} + \frac{x^{2}}{640000})]$

Step 4.2

Rewrite $- 1 \frac{x}{400}$ as $- \frac{x}{400}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{400} + \frac{x^{2}}{640000})]$

Step 5

Since $4000$ is constant with respect to $x$ , the derivative of $4000 (1 - \frac{x}{400} + \frac{x^{2}}{640000})$ with respect to $x$ is $4000 \frac{d}{d x} [1 - \frac{x}{400} + \frac{x^{2}}{640000}]$ .

$4000 \frac{d}{d x} [1 - \frac{x}{400} + \frac{x^{2}}{640000}]$

Step 6

By the Sum Rule, the derivative of $1 - \frac{x}{400} + \frac{x^{2}}{640000}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{400}] + \frac{d}{d x} [\frac{x^{2}}{640000}]$ .

$4000 (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{400}] + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 7

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

$4000 (0 + \frac{d}{d x} [- \frac{x}{400}] + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 8

Add $0$ and $\frac{d}{d x} [- \frac{x}{400}]$ .

$4000 (\frac{d}{d x} [- \frac{x}{400}] + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 9

Since $- \frac{1}{400}$ is constant with respect to $x$ , the derivative of $- \frac{x}{400}$ with respect to $x$ is $- \frac{1}{400} \frac{d}{d x} [x]$ .

$4000 (- \frac{1}{400} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 10

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

$4000 (- \frac{1}{400} \cdot 1 + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 11

Multiply $- 1$ by $1$ .

$4000 (- \frac{1}{400} + \frac{d}{d x} [\frac{x^{2}}{640000}])$

Step 12

Since $\frac{1}{640000}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{640000}$ with respect to $x$ is $\frac{1}{640000} \frac{d}{d x} [x^{2}]$ .

$4000 (- \frac{1}{400} + \frac{1}{640000} \frac{d}{d x} [x^{2}])$

Step 13

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

$4000 (- \frac{1}{400} + \frac{1}{640000} (2 x))$

Step 14

Simplify terms.

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Step 14.1

Combine $2$ and $\frac{1}{640000}$ .

$4000 (- \frac{1}{400} + \frac{2}{640000} x)$

Step 14.2

Combine $\frac{2}{640000}$ and $x$ .

$4000 (- \frac{1}{400} + \frac{2 x}{640000})$

Step 14.3

Cancel the common factor of $2$ and $640000$ .

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Step 14.3.1

Factor $2$ out of $2 x$ .

$4000 (- \frac{1}{400} + \frac{2 (x)}{640000})$

Step 14.3.2

Cancel the common factors.

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Step 14.3.2.1

Factor $2$ out of $640000$ .

$4000 (- \frac{1}{400} + \frac{2 x}{2 \cdot 320000})$

Step 14.3.2.2

Cancel the common factor.

$4000 (- \frac{1}{400} + \frac{2 x}{2 \cdot 320000})$

Step 14.3.2.3

Rewrite the expression.

$4000 (- \frac{1}{400} + \frac{x}{320000})$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$4000 (- \frac{1}{400}) + 4000 \frac{x}{320000}$

Step 15.2

Combine terms.

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Step 15.2.1

Multiply $- 1$ by $4000$ .

$- 4000 (\frac{1}{400}) + 4000 \frac{x}{320000}$

Step 15.2.2

Combine $- 4000$ and $\frac{1}{400}$ .

$\frac{- 4000}{400} + 4000 \frac{x}{320000}$

Step 15.2.3

Cancel the common factor of $- 4000$ and $400$ .

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Step 15.2.3.1

Factor $400$ out of $- 4000$ .

$\frac{400 \cdot - 10}{400} + 4000 \frac{x}{320000}$

Step 15.2.3.2

Cancel the common factors.

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Step 15.2.3.2.1

Factor $400$ out of $400$ .

$\frac{400 \cdot - 10}{400 (1)} + 4000 \frac{x}{320000}$

Step 15.2.3.2.2

Cancel the common factor.

$\frac{400 \cdot - 10}{400 \cdot 1} + 4000 \frac{x}{320000}$

Step 15.2.3.2.3

Rewrite the expression.

$\frac{- 10}{1} + 4000 \frac{x}{320000}$

Step 15.2.3.2.4

Divide $- 10$ by $1$ .

$- 10 + 4000 \frac{x}{320000}$

Step 15.2.4

Combine $4000$ and $\frac{x}{320000}$ .

$- 10 + \frac{4000 x}{320000}$

Step 15.2.5

Cancel the common factor of $4000$ and $320000$ .

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Step 15.2.5.1

Factor $4000$ out of $4000 x$ .

$- 10 + \frac{4000 (x)}{320000}$

Step 15.2.5.2

Cancel the common factors.

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Step 15.2.5.2.1

Factor $4000$ out of $320000$ .

$- 10 + \frac{4000 x}{4000 \cdot 80}$

Step 15.2.5.2.2

Cancel the common factor.

$- 10 + \frac{4000 x}{4000 \cdot 80}$

Step 15.2.5.2.3

Rewrite the expression.

$- 10 + \frac{x}{80}$

Step 15.3

Reorder terms.

$\frac{x}{80} - 10$