Calculus Examples

$P (x) = \frac{- {0.1}^{2} + 100 x - 60}{4000 x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Raise $0.1$ to the power of $2$ .

$\frac{d}{d x} [\frac{- 1 \cdot 0.01 + 100 x - 60}{4000 x}]$

Step 1.1.2

Simplify the expression.

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Step 1.1.2.1

Multiply $- 1$ by $0.01$ .

$\frac{d}{d x} [\frac{- 0.01 + 100 x - 60}{4000 x}]$

Step 1.1.2.2

Subtract $60$ from $- 0.01$ .

$\frac{d}{d x} [\frac{100 x - 60.01}{4000 x}]$

Step 1.1.3

Since $\frac{1}{4000}$ is constant with respect to $x$ , the derivative of $\frac{100 x - 60.01}{4000 x}$ with respect to $x$ is $\frac{1}{4000} \frac{d}{d x} [\frac{100 x - 60.01}{x}]$ .

$\frac{1}{4000} \frac{d}{d x} [\frac{100 x - 60.01}{x}]$

Step 1.2

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 $f (x) = 100 x - 60.01$ and $g (x) = x$ .

$\frac{1}{4000} \cdot \frac{x \frac{d}{d x} [100 x - 60.01] - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $100 x - 60.01$ with respect to $x$ is $\frac{d}{d x} [100 x] + \frac{d}{d x} [- 60.01]$ .

$\frac{1}{4000} \cdot \frac{x (\frac{d}{d x} [100 x] + \frac{d}{d x} [- 60.01]) - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.2

Since $100$ is constant with respect to $x$ , the derivative of $100 x$ with respect to $x$ is $100 \frac{d}{d x} [x]$ .

$\frac{1}{4000} \cdot \frac{x (100 \frac{d}{d x} [x] + \frac{d}{d x} [- 60.01]) - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3

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

$\frac{1}{4000} \cdot \frac{x (100 \cdot 1 + \frac{d}{d x} [- 60.01]) - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4

Multiply $100$ by $1$ .

$\frac{1}{4000} \cdot \frac{x (100 + \frac{d}{d x} [- 60.01]) - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.5

Since $- 60.01$ is constant with respect to $x$ , the derivative of $- 60.01$ with respect to $x$ is $0$ .

$\frac{1}{4000} \cdot \frac{x (100 + 0) - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $100$ and $0$ .

$\frac{1}{4000} \cdot \frac{x \cdot 100 - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.6.2

Move $100$ to the left of $x$ .

$\frac{1}{4000} \cdot \frac{100 \cdot x - (100 x - 60.01) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.7

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

$\frac{1}{4000} \cdot \frac{100 x - (100 x - 60.01) \cdot 1}{x^{2}}$

Step 1.3.8

Combine fractions.

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Step 1.3.8.1

Multiply $- 1$ by $1$ .

$\frac{1}{4000} \cdot \frac{100 x - (100 x - 60.01)}{x^{2}}$

Step 1.3.8.2

Multiply $\frac{1}{4000}$ by $\frac{100 x - (100 x - 60.01)}{x^{2}}$ .

$\frac{100 x - (100 x - 60.01)}{4000 x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$\frac{100 x - (100 x) - - 60.01}{4000 x^{2}}$

Step 1.4.2

Simplify the numerator.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Multiply $100$ by $- 1$ .

$\frac{100 x - 100 x - - 60.01}{4000 x^{2}}$

Step 1.4.2.1.2

Multiply $- 1$ by $- 60.01$ .

$\frac{100 x - 100 x + 60.01}{4000 x^{2}}$

Step 1.4.2.2

Subtract $100 x$ from $100 x$ .

$\frac{0 + 60.01}{4000 x^{2}}$

Step 1.4.2.3

Add $0$ and $60.01$ .

$\frac{60.01}{4000 x^{2}}$

Step 1.4.3

Factor $60.01$ out of $60.01$ .

$\frac{60.01 (1)}{4000 x^{2}}$

Step 1.4.4

Factor $4000$ out of $4000 x^{2}$ .

$\frac{60.01 (1)}{4000 (x^{2})}$

Step 1.4.5

Separate fractions.

$\frac{60.01}{4000} \cdot \frac{1}{x^{2}}$

Step 1.4.6

Divide $60.01$ by $4000$ .

$0.0150025 \frac{1}{x^{2}}$

Step 1.4.7

Combine $0.0150025$ and $\frac{1}{x^{2}}$ .

$\frac{0.0150025}{x^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $0.0150025$ is constant with respect to $x$ , the derivative of $\frac{0.0150025}{x^{2}}$ with respect to $x$ is $0.0150025 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$P'' (x) = 0.0150025 \frac{d}{d x} (\frac{1}{x^{2}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

$P'' (x) = 0.0150025 \frac{d}{d x} ({(x^{2})}^{- 1})$

Step 2.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$P'' (x) = 0.0150025 \frac{d}{d x} (x^{2 \cdot - 1})$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$P'' (x) = 0.0150025 \frac{d}{d x} (x^{- 2})$

Step 2.3

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

$P'' (x) = 0.0150025 (- 2 x^{- 3})$

Step 2.4

Multiply $- 2$ by $0.0150025$ .

$P'' (x) = - 0.030005 x^{- 3}$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$P'' (x) = - 0.030005 \frac{1}{x^{3}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 0.030005$ and $\frac{1}{x^{3}}$ .

$P'' (x) = \frac{- 0.030005}{x^{3}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$P'' (x) = - \frac{0.030005}{x^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{0.0150025}{x^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6