Calculus Examples

$\frac{\frac{10 - 0.0002 x}{x}}{- 0.0002}$

Step 1

Rewrite $\frac{\frac{10 - 0.0002 x}{x}}{- 0.0002}$ as a product.

$\frac{d}{d x} [\frac{10 - 0.0002 x}{x} \cdot \frac{1}{- 0.0002}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Move the negative in front of the fraction.

$\frac{d}{d x} [\frac{10 - 0.0002 x}{x} (- \frac{1}{0.0002})]$

Step 2.2

Combine fractions.

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Step 2.2.1

Multiply $\frac{10 - 0.0002 x}{x}$ by $\frac{1}{0.0002}$ .

$\frac{d}{d x} [- \frac{10 - 0.0002 x}{x \cdot 0.0002}]$

Step 2.2.2

Move $0.0002$ to the left of $x$ .

$\frac{d}{d x} [- \frac{10 - 0.0002 x}{0.0002 \cdot x}]$

Step 2.3

Since $- \frac{1}{0.0002}$ is constant with respect to $x$ , the derivative of $- \frac{10 - 0.0002 x}{0.0002 x}$ with respect to $x$ is $- \frac{1}{0.0002} \frac{d}{d x} [\frac{10 - 0.0002 x}{x}]$ .

$- \frac{1}{0.0002} \frac{d}{d x} [\frac{10 - 0.0002 x}{x}]$

Step 3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 10 - 0.0002 x$ and $g (x) = x$ .

$- \frac{1}{0.0002} \cdot \frac{x \frac{d}{d x} [10 - 0.0002 x] - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $10 - 0.0002 x$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- 0.0002 x]$ .

$- \frac{1}{0.0002} \cdot \frac{x (\frac{d}{d x} [10] + \frac{d}{d x} [- 0.0002 x]) - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$- \frac{1}{0.0002} \cdot \frac{x (0 + \frac{d}{d x} [- 0.0002 x]) - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

Add $0$ and $\frac{d}{d x} [- 0.0002 x]$ .

$- \frac{1}{0.0002} \cdot \frac{x \frac{d}{d x} [- 0.0002 x] - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.4

Since $- 0.0002$ is constant with respect to $x$ , the derivative of $- 0.0002 x$ with respect to $x$ is $- 0.0002 \frac{d}{d x} [x]$ .

$- \frac{1}{0.0002} \cdot \frac{x (- 0.0002 \frac{d}{d x} [x]) - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.5

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

$- \frac{1}{0.0002} \cdot \frac{x (- 0.0002 \cdot 1) - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Multiply $- 0.0002$ by $1$ .

$- \frac{1}{0.0002} \cdot \frac{x \cdot - 0.0002 - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.6.2

Move $- 0.0002$ to the left of $x$ .

$- \frac{1}{0.0002} \cdot \frac{- 0.0002 \cdot x - (10 - 0.0002 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.7

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

$- \frac{1}{0.0002} \cdot \frac{- 0.0002 x - (10 - 0.0002 x) \cdot 1}{x^{2}}$

Step 4.8

Combine fractions.

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Step 4.8.1

Multiply $- 1$ by $1$ .

$- \frac{1}{0.0002} \cdot \frac{- 0.0002 x - (10 - 0.0002 x)}{x^{2}}$

Step 4.8.2

Multiply $\frac{- 0.0002 x - (10 - 0.0002 x)}{x^{2}}$ by $\frac{1}{0.0002}$ .

$- \frac{- 0.0002 x - (10 - 0.0002 x)}{x^{2} \cdot 0.0002}$

Step 4.8.3

Move $0.0002$ to the left of $x^{2}$ .

$- \frac{- 0.0002 x - (10 - 0.0002 x)}{0.0002 \cdot x^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{- 0.0002 x - 1 \cdot 10 - (- 0.0002 x)}{0.0002 x^{2}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 1$ by $10$ .

$- \frac{- 0.0002 x - 10 - (- 0.0002 x)}{0.0002 x^{2}}$

Step 5.2.1.2

Multiply $- 0.0002$ by $- 1$ .

$- \frac{- 0.0002 x - 10 + 0.0002 x}{0.0002 x^{2}}$

Step 5.2.2

Add $- 0.0002 x$ and $0.0002 x$ .

$- \frac{0 x - 10}{0.0002 x^{2}}$

Step 5.2.3

Multiply $0$ by $x$ .

$- \frac{0 - 10}{0.0002 x^{2}}$

Step 5.2.4

Subtract $10$ from $0$ .

$- \frac{- 10}{0.0002 x^{2}}$

Step 5.3

Combine terms.

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Step 5.3.1

Move the negative in front of the fraction.

$- - \frac{10}{0.0002 x^{2}}$

Step 5.3.2

Multiply $- 1$ by $- 1$ .

$1 \frac{10}{0.0002 x^{2}}$

Step 5.3.3

Multiply $\frac{10}{0.0002 x^{2}}$ by $1$ .

$\frac{10}{0.0002 x^{2}}$

Step 5.4

Factor $10$ out of $10$ .

$\frac{10 (1)}{0.0002 x^{2}}$

Step 5.5

Factor $0.0002$ out of $0.0002 x^{2}$ .

$\frac{10 (1)}{0.0002 (x^{2})}$

Step 5.6

Separate fractions.

$\frac{10}{0.0002} \cdot \frac{1}{x^{2}}$

Step 5.7

Divide $10$ by $0.0002$ .

$50000 \frac{1}{x^{2}}$

Step 5.8

Combine $50000$ and $\frac{1}{x^{2}}$ .

$\frac{50000}{x^{2}}$