Calculus Examples

$f (x) = 4 p x^{2} + \frac{804.248}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 p x^{2} + \frac{804.248}{x}$ with respect to $x$ is $\frac{d}{d x} [4 p x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$ .

$\frac{d}{d x} [4 p x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [4 p x^{2}]$ .

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Step 1.2.1

Since $4 p$ is constant with respect to $x$ , the derivative of $4 p x^{2}$ with respect to $x$ is $4 p \frac{d}{d x} [x^{2}]$ .

$4 p \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$

Step 1.2.2

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

$4 p (2 x) + \frac{d}{d x} [\frac{804.248}{x}]$

Step 1.2.3

Multiply $2$ by $4$ .

$8 p x + \frac{d}{d x} [\frac{804.248}{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{804.248}{x}]$ .

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Step 1.3.1

Since $804.248$ is constant with respect to $x$ , the derivative of $\frac{804.248}{x}$ with respect to $x$ is $804.248 \frac{d}{d x} [\frac{1}{x}]$ .

$8 p x + 804.248 \frac{d}{d x} [\frac{1}{x}]$

Step 1.3.2

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

$8 p x + 804.248 \frac{d}{d x} [x^{- 1}]$

Step 1.3.3

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

$8 p x + 804.248 (- x^{- 2})$

Step 1.3.4

Multiply $- 1$ by $804.248$ .

$8 p x - 804.248 x^{- 2}$

Step 1.4

Simplify.

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Step 1.4.1

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

$8 p x - 804.248 \frac{1}{x^{2}}$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $- 804.248$ and $\frac{1}{x^{2}}$ .

$8 p x + \frac{- 804.248}{x^{2}}$

Step 1.4.2.2

Move the negative in front of the fraction.

$8 p x - \frac{804.248}{x^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $8 p x - \frac{804.248}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [8 p x] + \frac{d}{d x} [- \frac{804.248}{x^{2}}]$ .

$f'' (x) = \frac{d}{d x} (8 p x) + \frac{d}{d x} (- \frac{804.248}{x^{2}})$

Step 2.2

Evaluate $\frac{d}{d x} [8 p x]$ .

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Step 2.2.1

Since $8 p$ is constant with respect to $x$ , the derivative of $8 p x$ with respect to $x$ is $8 p \frac{d}{d x} [x]$ .

$f'' (x) = 8 p \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{804.248}{x^{2}})$

Step 2.2.2

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

$f'' (x) = 8 p \cdot 1 + \frac{d}{d x} (- \frac{804.248}{x^{2}})$

Step 2.2.3

Multiply $8$ by $1$ .

$f'' (x) = 8 p + \frac{d}{d x} (- \frac{804.248}{x^{2}})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{804.248}{x^{2}}]$ .

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Step 2.3.1

Since $- 804.248$ is constant with respect to $x$ , the derivative of $- \frac{804.248}{x^{2}}$ with respect to $x$ is $- 804.248 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = 8 p - 804.248 \frac{d}{d x} \frac{1}{x^{2}}$

Step 2.3.2

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

$f'' (x) = 8 p - 804.248 \frac{d}{d x} {(x^{2})}^{- 1}$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$f'' (x) = 8 p - 804.248 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2}))$

Step 2.3.3.2

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

$f'' (x) = 8 p - 804.248 (- u^{- 2} \frac{d}{d x} x^{2})$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$f'' (x) = 8 p - 804.248 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2})$

Step 2.3.4

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

$f'' (x) = 8 p - 804.248 (- {(x^{2})}^{- 2} (2 x))$

Step 2.3.5

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

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Step 2.3.5.1

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

$f'' (x) = 8 p - 804.248 (- x^{2 \cdot - 2} (2 x))$

Step 2.3.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = 8 p - 804.248 (- x^{- 4} (2 x))$

Step 2.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = 8 p - 804.248 (- 2 x^{- 4} x)$

Step 2.3.7

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

$f'' (x) = 8 p - 804.248 (- 2 (x \cdot x^{- 4}))$

Step 2.3.8

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

$f'' (x) = 8 p - 804.248 (- 2 x^{1 - 4})$

Step 2.3.9

Subtract $4$ from $1$ .

$f'' (x) = 8 p - 804.248 (- 2 x^{- 3})$

Step 2.3.10

Multiply $- 2$ by $- 804.248$ .

$f'' (x) = 8 p + 1608.496 x^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = 8 p + 1608.496 (\frac{1}{x^{3}})$

Step 2.4.2

Combine $1608.496$ and $\frac{1}{x^{3}}$ .

$f'' (x) = 8 p + \frac{1608.496}{x^{3}}$

Step 3

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

$8 p x - \frac{804.248}{x^{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $4 p x^{2} + \frac{804.248}{x}$ with respect to $x$ is $\frac{d}{d x} [4 p x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$ .

$\frac{d}{d x} [4 p x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [4 p x^{2}]$ .

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Step 4.1.2.1

Since $4 p$ is constant with respect to $x$ , the derivative of $4 p x^{2}$ with respect to $x$ is $4 p \frac{d}{d x} [x^{2}]$ .

$4 p \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{804.248}{x}]$

Step 4.1.2.2

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

$4 p (2 x) + \frac{d}{d x} [\frac{804.248}{x}]$

Step 4.1.2.3

Multiply $2$ by $4$ .

$8 p x + \frac{d}{d x} [\frac{804.248}{x}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{804.248}{x}]$ .

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Step 4.1.3.1

Since $804.248$ is constant with respect to $x$ , the derivative of $\frac{804.248}{x}$ with respect to $x$ is $804.248 \frac{d}{d x} [\frac{1}{x}]$ .

$8 p x + 804.248 \frac{d}{d x} [\frac{1}{x}]$

Step 4.1.3.2

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

$8 p x + 804.248 \frac{d}{d x} [x^{- 1}]$

Step 4.1.3.3

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

$8 p x + 804.248 (- x^{- 2})$

Step 4.1.3.4

Multiply $- 1$ by $804.248$ .

$8 p x - 804.248 x^{- 2}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

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

$8 p x - 804.248 \frac{1}{x^{2}}$

Step 4.1.4.2

Combine terms.

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Step 4.1.4.2.1

Combine $- 804.248$ and $\frac{1}{x^{2}}$ .

$8 p x + \frac{- 804.248}{x^{2}}$

Step 4.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = 8 p x - \frac{804.248}{x^{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $8 p x - \frac{804.248}{x^{2}}$ .

$8 p x - \frac{804.248}{x^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $8 p x - \frac{804.248}{x^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$8 p x - \frac{804.248}{x^{2}} = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5.3

Multiply each term in $8 p x - \frac{804.248}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $8 p x - \frac{804.248}{x^{2}} = 0$ by $x^{2}$ .

$8 p x \cdot x^{2} - \frac{804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.3.2.1.1.1

Move $x^{2}$ .

$8 p (x^{2} x) - \frac{804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 5.3.2.1.1.2.1

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

$8 p (x^{2} x^{1}) - \frac{804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.1.2.2

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

$8 p x^{2 + 1} - \frac{804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.1.3

Add $2$ and $1$ .

$8 p x^{3} - \frac{804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{804.248}{x^{2}}$ into the numerator.

$8 p x^{3} + \frac{- 804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.2.2

Cancel the common factor.

$8 p x^{3} + \frac{- 804.248}{x^{2}} x^{2} = 0 x^{2}$

Step 5.3.2.1.2.3

Rewrite the expression.

$8 p x^{3} - 804.248 = 0 x^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x^{2}$ .

$8 p x^{3} - 804.248 = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Add $804.248$ to both sides of the equation.

$8 p x^{3} = 804.248$

Step 5.4.2

Divide each term in $8 p x^{3} = 804.248$ by $8 p$ and simplify.

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Step 5.4.2.1

Divide each term in $8 p x^{3} = 804.248$ by $8 p$ .

$\frac{8 p x^{3}}{8 p} = \frac{804.248}{8 p}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $8$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{8 p x^{3}}{8 p} = \frac{804.248}{8 p}$

Step 5.4.2.2.1.2

Rewrite the expression.

$\frac{p x^{3}}{p} = \frac{804.248}{8 p}$

Step 5.4.2.2.2

Cancel the common factor of $p$ .

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Step 5.4.2.2.2.1

Cancel the common factor.

$\frac{p x^{3}}{p} = \frac{804.248}{8 p}$

Step 5.4.2.2.2.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{804.248}{8 p}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Factor $804.248$ out of $804.248$ .

$x^{3} = \frac{804.248 (1)}{8 p}$

Step 5.4.2.3.2

Factor $8$ out of $8 p$ .

$x^{3} = \frac{804.248 (1)}{8 (p)}$

Step 5.4.2.3.3

Separate fractions.

$x^{3} = \frac{804.248}{8} \cdot \frac{1}{p}$

Step 5.4.2.3.4

Divide $804.248$ by $8$ .

$x^{3} = 100.531 \frac{1}{p}$

Step 5.4.2.3.5

Combine $100.531$ and $\frac{1}{p}$ .

$x^{3} = \frac{100.531}{p}$

Step 5.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{100.531}{p}}$

Step 5.4.4

Simplify $\sqrt[3]{\frac{100.531}{p}}$ .

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Step 5.4.4.1

Rewrite $\sqrt[3]{\frac{100.531}{p}}$ as $\frac{\sqrt[3]{100.531}}{\sqrt[3]{p}}$ .

$x = \frac{\sqrt[3]{100.531}}{\sqrt[3]{p}}$

Step 5.4.4.2

Multiply $\frac{\sqrt[3]{100.531}}{\sqrt[3]{p}}$ by $\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$x = \frac{\sqrt[3]{100.531}}{\sqrt[3]{p}} \cdot \frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$

Step 5.4.4.3

Combine and simplify the denominator.

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Step 5.4.4.3.1

Multiply $\frac{\sqrt[3]{100.531}}{\sqrt[3]{p}}$ by $\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{\sqrt[3]{p} {\sqrt[3]{p}}^{2}}$

Step 5.4.4.3.2

Raise $\sqrt[3]{p}$ to the power of $1$ .

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1} {\sqrt[3]{p}}^{2}}$

Step 5.4.4.3.3

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

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1 + 2}}$

Step 5.4.4.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{3}}$

Step 5.4.4.3.5

Rewrite ${\sqrt[3]{p}}^{3}$ as $p$ .

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Step 5.4.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{p}$ as $p^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{{(p^{\frac{1}{3}})}^{3}}$

Step 5.4.4.3.5.2

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

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{p^{\frac{1}{3} \cdot 3}}$

Step 5.4.4.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 5.4.4.3.5.4

Cancel the common factor of $3$ .

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Step 5.4.4.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 5.4.4.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{p^{1}}$

Step 5.4.4.3.5.5

Simplify.

$x = \frac{\sqrt[3]{100.531} {\sqrt[3]{p}}^{2}}{p}$

Step 5.4.4.4

Rewrite ${\sqrt[3]{p}}^{2}$ as $\sqrt[3]{p^{2}}$ .

$x = \frac{\sqrt[3]{100.531} \sqrt[3]{p^{2}}}{p}$

Step 5.4.4.5

Combine using the product rule for radicals.

$x = \frac{\sqrt[3]{100.531 p^{2}}}{p}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{804.248}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.2

Solve for $p$ .

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Step 6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$p = 0$

Step 7

Critical points to evaluate.

$x = \frac{\sqrt[3]{100.531 p^{2}}}{p}$

Step 8

Evaluate the second derivative at $x = \frac{\sqrt[3]{100.531 p^{2}}}{p}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$8 p + \frac{1608.496}{{(\frac{\sqrt[3]{100.531 p^{2}}}{p})}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Apply the product rule to $\frac{\sqrt[3]{100.531 p^{2}}}{p}$ .

$8 p + \frac{1608.496}{\frac{{\sqrt[3]{100.531 p^{2}}}^{3}}{p^{3}}}$

Step 9.1.1.2

Rewrite ${\sqrt[3]{100.531 p^{2}}}^{3}$ as $100.531 p^{2}$ .

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Step 9.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{100.531 p^{2}}$ as ${(100.531 p^{2})}^{\frac{1}{3}}$ .

$8 p + \frac{1608.496}{\frac{{({(100.531 p^{2})}^{\frac{1}{3}})}^{3}}{p^{3}}}$

Step 9.1.1.2.2

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

$8 p + \frac{1608.496}{\frac{{(100.531 p^{2})}^{\frac{1}{3} \cdot 3}}{p^{3}}}$

Step 9.1.1.2.3

Combine $\frac{1}{3}$ and $3$ .

$8 p + \frac{1608.496}{\frac{{(100.531 p^{2})}^{\frac{3}{3}}}{p^{3}}}$

Step 9.1.1.2.4

Cancel the common factor of $3$ .

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Step 9.1.1.2.4.1

Cancel the common factor.

$8 p + \frac{1608.496}{\frac{{(100.531 p^{2})}^{\frac{3}{3}}}{p^{3}}}$

Step 9.1.1.2.4.2

Rewrite the expression.

$8 p + \frac{1608.496}{\frac{{(100.531 p^{2})}^{1}}{p^{3}}}$

Step 9.1.1.2.5

Simplify.

$8 p + \frac{1608.496}{\frac{100.531 p^{2}}{p^{3}}}$

Step 9.1.1.3

Cancel the common factor of $p^{2}$ and $p^{3}$ .

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Step 9.1.1.3.1

Factor $p^{2}$ out of $100.531 p^{2}$ .

$8 p + \frac{1608.496}{\frac{p^{2} \cdot 100.531}{p^{3}}}$

Step 9.1.1.3.2

Cancel the common factors.

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Step 9.1.1.3.2.1

Factor $p^{2}$ out of $p^{3}$ .

$8 p + \frac{1608.496}{\frac{p^{2} \cdot 100.531}{p^{2} p}}$

Step 9.1.1.3.2.2

Cancel the common factor.

$8 p + \frac{1608.496}{\frac{p^{2} \cdot 100.531}{p^{2} p}}$

Step 9.1.1.3.2.3

Rewrite the expression.

$8 p + \frac{1608.496}{\frac{100.531}{p}}$

Step 9.1.2

Multiply the numerator by the reciprocal of the denominator.

$8 p + 1608.496 \frac{p}{100.531}$

Step 9.1.3

Cancel the common factor of $100.531$ .

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Step 9.1.3.1

Factor $100.531$ out of $1608.496$ .

$8 p + 100.531 (16) \frac{p}{100.531}$

Step 9.1.3.2

Cancel the common factor.

$8 p + 100.531 \cdot 16 \frac{p}{100.531}$

Step 9.1.3.3

Rewrite the expression.

$8 p + 16 p$

Step 9.2

Add $8 p$ and $16 p$ .

$24 p$

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11