Calculus Examples

$g (x) = \frac{1}{115 \sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})$

Step 1

Find the first derivative of the function.

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Step 1.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{d}{d x} [\frac{1}{115 \sqrt{2 p}} \cdot \frac{\sqrt{2 p}}{\sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 \sqrt{2 p} \sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.2

Move $\sqrt{2 p}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 (\sqrt{2 p} \sqrt{2 p})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.3

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} \sqrt{2 p})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.4

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} {\sqrt{2 p}}^{1})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.5

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{1 + 1}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.6

Add $1$ and $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7

Rewrite ${\sqrt{2 p}}^{2}$ as $2 p$ .

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Step 1.2.7.1

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {({(2 p)}^{\frac{1}{2}})}^{2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7.2

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{1}{2} \cdot 2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7.3

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7.4.2

Rewrite the expression.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{1}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.2.7.5

Simplify.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 (2 p)} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.3

Multiply $115$ by $2$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 1.4

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} {(\frac{x - 512}{115})}^{2})]$

Step 1.5

Differentiate using the Constant Multiple Rule.

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Step 1.5.1

Apply the product rule to $\frac{x - 512}{115}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} \cdot \frac{{(x - 512)}^{2}}{115^{2}})]$

Step 1.5.2

Combine fractions.

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Step 1.5.2.1

Raise $115$ to the power of $2$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} \cdot \frac{{(x - 512)}^{2}}{13225})]$

Step 1.5.2.2

Multiply $\frac{1}{e^{\frac{1}{2}}}$ by $\frac{{(x - 512)}^{2}}{13225}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot \frac{{(x - 512)}^{2}}{e^{\frac{1}{2}} \cdot 13225}]$

Step 1.5.2.3

Move $13225$ to the left of $e^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot \frac{{(x - 512)}^{2}}{13225 \cdot e^{\frac{1}{2}}}]$

Step 1.5.2.4

Multiply $\frac{\sqrt{2 p}}{230 p}$ by $\frac{{(x - 512)}^{2}}{13225 e^{\frac{1}{2}}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p} {(x - 512)}^{2}}{230 p (13225 e^{\frac{1}{2}})}]$

Step 1.5.2.5

Multiply $13225$ by $230$ .

$\frac{d}{d x} [\frac{\sqrt{2 p} {(x - 512)}^{2}}{3041750 p e^{\frac{1}{2}}}]$

Step 1.5.3

Since $\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt{2 p} {(x - 512)}^{2}}{3041750 p e^{\frac{1}{2}}}$ with respect to $x$ is $\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} \frac{d}{d x} [{(x - 512)}^{2}]$ .

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} \frac{d}{d x} [{(x - 512)}^{2}]$

Step 1.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 512$ .

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Step 1.6.1

To apply the Chain Rule, set $u$ as $x - 512$ .

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 512])$

Step 1.6.2

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

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (2 u \frac{d}{d x} [x - 512])$

Step 1.6.3

Replace all occurrences of $u$ with $x - 512$ .

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (2 (x - 512) \frac{d}{d x} [x - 512])$

Step 1.7

Simplify terms.

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Step 1.7.1

Combine $2$ and $\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}}$ .

$\frac{2 \sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 1.7.2

Factor $2$ out of $2 \sqrt{2 p}$ .

$\frac{2 (\sqrt{2 p})}{3041750 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 1.8

Cancel the common factors.

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Step 1.8.1

Factor $2$ out of $3041750 p e^{\frac{1}{2}}$ .

$\frac{2 (\sqrt{2 p})}{2 (1520875 p e^{\frac{1}{2}})} ((x - 512) \frac{d}{d x} [x - 512])$

Step 1.8.2

Cancel the common factor.

$\frac{2 \sqrt{2 p}}{2 (1520875 p e^{\frac{1}{2}})} ((x - 512) \frac{d}{d x} [x - 512])$

Step 1.8.3

Rewrite the expression.

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 1.9

By the Sum Rule, the derivative of $x - 512$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 512]$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (\frac{d}{d x} [x] + \frac{d}{d x} [- 512])$

Step 1.10

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

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (1 + \frac{d}{d x} [- 512])$

Step 1.11

Since $- 512$ is constant with respect to $x$ , the derivative of $- 512$ with respect to $x$ is $0$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (1 + 0)$

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) \cdot 1$

Step 1.12.2

Multiply $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ by $1$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512)$

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} x + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot - 512$

Step 1.13.2

Combine terms.

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Step 1.13.2.1

Combine $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ and $x$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot - 512$

Step 1.13.2.2

Combine $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ and $- 512$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} \cdot - 512}{1520875 p e^{\frac{1}{2}}}$

Step 1.13.2.3

Move $- 512$ to the left of $\sqrt{2 p}$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{- 512 \cdot \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 1.13.2.4

Move the negative in front of the fraction.

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} - \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 1.13.3

Reorder terms.

$- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}]$ .

$g'' (x) = \frac{d}{d x} (- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}) + \frac{d}{d x} (\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}})$

Step 2.1.2

Since $- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ is constant with respect to $x$ , the derivative of $- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ with respect to $x$ is $0$ .

$g'' (x) = 0 + \frac{d}{d x} (\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}]$ .

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Step 2.2.1

Since $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$ with respect to $x$ is $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \frac{d}{d x} [x]$ .

$g'' (x) = 0 + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot \frac{d}{d x} (x)$

Step 2.2.2

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

$g'' (x) = 0 + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot 1$

Step 2.2.3

Multiply $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ by $1$ .

$g'' (x) = 0 + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 2.3

Simplify.

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Step 2.3.1

Add $0$ and $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ .

$g'' (x) = \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 2.3.2

Reorder terms.

$g'' (x) = \frac{\sqrt{2 p}}{1520875 e^{\frac{1}{2}} p}$

Step 2.3.3

Reorder factors in $\frac{\sqrt{2 p}}{1520875 e^{\frac{1}{2}} p}$ .

$g'' (x) = \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 3

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

$- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} = 0$

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

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{d}{d x} [\frac{1}{115 \sqrt{2 p}} \cdot \frac{\sqrt{2 p}}{\sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2

Combine and simplify the denominator.

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Step 4.1.2.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 \sqrt{2 p} \sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.2

Move $\sqrt{2 p}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 (\sqrt{2 p} \sqrt{2 p})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.3

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} \sqrt{2 p})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.4

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} {\sqrt{2 p}}^{1})} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.5

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{1 + 1}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.6

Add $1$ and $1$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7

Rewrite ${\sqrt{2 p}}^{2}$ as $2 p$ .

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Step 4.1.2.7.1

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {({(2 p)}^{\frac{1}{2}})}^{2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7.2

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{1}{2} \cdot 2}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7.3

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7.4

Cancel the common factor of $2$ .

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Step 4.1.2.7.4.1

Cancel the common factor.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7.4.2

Rewrite the expression.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 {(2 p)}^{1}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.2.7.5

Simplify.

$\frac{d}{d x} [\frac{\sqrt{2 p}}{115 (2 p)} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.3

Multiply $115$ by $2$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.4

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

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} {(\frac{x - 512}{115})}^{2})]$

Step 4.1.5

Differentiate using the Constant Multiple Rule.

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Step 4.1.5.1

Apply the product rule to $\frac{x - 512}{115}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} \cdot \frac{{(x - 512)}^{2}}{115^{2}})]$

Step 4.1.5.2

Combine fractions.

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Step 4.1.5.2.1

Raise $115$ to the power of $2$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot (\frac{1}{e^{\frac{1}{2}}} \cdot \frac{{(x - 512)}^{2}}{13225})]$

Step 4.1.5.2.2

Multiply $\frac{1}{e^{\frac{1}{2}}}$ by $\frac{{(x - 512)}^{2}}{13225}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot \frac{{(x - 512)}^{2}}{e^{\frac{1}{2}} \cdot 13225}]$

Step 4.1.5.2.3

Move $13225$ to the left of $e^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p}}{230 p} \cdot \frac{{(x - 512)}^{2}}{13225 \cdot e^{\frac{1}{2}}}]$

Step 4.1.5.2.4

Multiply $\frac{\sqrt{2 p}}{230 p}$ by $\frac{{(x - 512)}^{2}}{13225 e^{\frac{1}{2}}}$ .

$\frac{d}{d x} [\frac{\sqrt{2 p} {(x - 512)}^{2}}{230 p (13225 e^{\frac{1}{2}})}]$

Step 4.1.5.2.5

Multiply $13225$ by $230$ .

$\frac{d}{d x} [\frac{\sqrt{2 p} {(x - 512)}^{2}}{3041750 p e^{\frac{1}{2}}}]$

Step 4.1.5.3

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} \frac{d}{d x} [{(x - 512)}^{2}]$

Step 4.1.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 512$ .

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Step 4.1.6.1

To apply the Chain Rule, set $u$ as $x - 512$ .

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 512])$

Step 4.1.6.2

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

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (2 u \frac{d}{d x} [x - 512])$

Step 4.1.6.3

Replace all occurrences of $u$ with $x - 512$ .

$\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} (2 (x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.7

Simplify terms.

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Step 4.1.7.1

Combine $2$ and $\frac{\sqrt{2 p}}{3041750 p e^{\frac{1}{2}}}$ .

$\frac{2 \sqrt{2 p}}{3041750 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.7.2

Factor $2$ out of $2 \sqrt{2 p}$ .

$\frac{2 (\sqrt{2 p})}{3041750 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.8

Cancel the common factors.

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Step 4.1.8.1

Factor $2$ out of $3041750 p e^{\frac{1}{2}}$ .

$\frac{2 (\sqrt{2 p})}{2 (1520875 p e^{\frac{1}{2}})} ((x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.8.2

Cancel the common factor.

$\frac{2 \sqrt{2 p}}{2 (1520875 p e^{\frac{1}{2}})} ((x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.8.3

Rewrite the expression.

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} ((x - 512) \frac{d}{d x} [x - 512])$

Step 4.1.9

By the Sum Rule, the derivative of $x - 512$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 512]$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (\frac{d}{d x} [x] + \frac{d}{d x} [- 512])$

Step 4.1.10

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

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (1 + \frac{d}{d x} [- 512])$

Step 4.1.11

Since $- 512$ is constant with respect to $x$ , the derivative of $- 512$ with respect to $x$ is $0$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) (1 + 0)$

Step 4.1.12

Simplify the expression.

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Step 4.1.12.1

Add $1$ and $0$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512) \cdot 1$

Step 4.1.12.2

Multiply $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ by $1$ .

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} (x - 512)$

Step 4.1.13

Simplify.

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Step 4.1.13.1

Apply the distributive property.

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} x + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot - 512$

Step 4.1.13.2

Combine terms.

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Step 4.1.13.2.1

Combine $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ and $x$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} \cdot - 512$

Step 4.1.13.2.2

Combine $\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ and $- 512$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} \cdot - 512}{1520875 p e^{\frac{1}{2}}}$

Step 4.1.13.2.3

Move $- 512$ to the left of $\sqrt{2 p}$ .

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} + \frac{- 512 \cdot \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 4.1.13.2.4

Move the negative in front of the fraction.

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} - \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 4.1.13.3

Reorder terms.

$f' (x) = - \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$

Step 4.2

The first derivative of $g (x)$ with respect to $x$ is $- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$ .

$- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} = 0$

Step 5.2

Add $\frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$ to both sides of the equation.

$\frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}} = \frac{512 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 5.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sqrt{2 p} x = 512 \sqrt{2 p}$

Step 5.4

Divide each term in $\sqrt{2 p} x = 512 \sqrt{2 p}$ by $\sqrt{2 p}$ and simplify.

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Step 5.4.1

Divide each term in $\sqrt{2 p} x = 512 \sqrt{2 p}$ by $\sqrt{2 p}$ .

$\frac{\sqrt{2 p} x}{\sqrt{2 p}} = \frac{512 \sqrt{2 p}}{\sqrt{2 p}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $\sqrt{2 p}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{\sqrt{2 p} x}{\sqrt{2 p}} = \frac{512 \sqrt{2 p}}{\sqrt{2 p}}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{512 \sqrt{2 p}}{\sqrt{2 p}}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $\sqrt{2 p}$ .

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Step 5.4.3.1.1

Cancel the common factor.

$x = \frac{512 \sqrt{2 p}}{\sqrt{2 p}}$

Step 5.4.3.1.2

Divide $512$ by $1$ .

$x = 512$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{512 \sqrt{2 p}}{1520875 p \sqrt{e^{1}}} + \frac{\sqrt{2 p} x}{1520875 p e^{\frac{1}{2}}}$

Step 6.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{512 \sqrt{2 p}}{1520875 p \sqrt{e^{1}}} + \frac{\sqrt{2 p} x}{1520875 p \sqrt{e^{1}}}$

Step 6.1.3

Anything raised to $1$ is the base itself.

$- \frac{512 \sqrt{2 p}}{1520875 p \sqrt{e}} + \frac{\sqrt{2 p} x}{1520875 p \sqrt{e^{1}}}$

Step 6.1.4

Anything raised to $1$ is the base itself.

$- \frac{512 \sqrt{2 p}}{1520875 p \sqrt{e}} + \frac{\sqrt{2 p} x}{1520875 p \sqrt{e}}$

Step 6.2

Set the denominator in $\frac{512 \sqrt{2 p}}{1520875 p \sqrt{e}}$ equal to $0$ to find where the expression is undefined.

$1520875 p \sqrt{e} = 0$

Step 6.3

Divide each term in $1520875 p \sqrt{e} = 0$ by $1520875 \sqrt{e}$ and simplify.

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Step 6.3.1

Divide each term in $1520875 p \sqrt{e} = 0$ by $1520875 \sqrt{e}$ .

$\frac{1520875 p \sqrt{e}}{1520875 \sqrt{e}} = \frac{0}{1520875 \sqrt{e}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $1520875$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{1520875 p \sqrt{e}}{1520875 \sqrt{e}} = \frac{0}{1520875 \sqrt{e}}$

Step 6.3.2.1.2

Rewrite the expression.

$\frac{p \sqrt{e}}{\sqrt{e}} = \frac{0}{1520875 \sqrt{e}}$

Step 6.3.2.2

Cancel the common factor of $\sqrt{e}$ .

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Step 6.3.2.2.1

Cancel the common factor.

$\frac{p \sqrt{e}}{\sqrt{e}} = \frac{0}{1520875 \sqrt{e}}$

Step 6.3.2.2.2

Divide $p$ by $1$ .

$p = \frac{0}{1520875 \sqrt{e}}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Cancel the common factor of $0$ and $1520875$ .

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Step 6.3.3.1.1

Factor $1520875$ out of $0$ .

$p = \frac{1520875 (0)}{1520875 \sqrt{e}}$

Step 6.3.3.1.2

Cancel the common factors.

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Step 6.3.3.1.2.1

Factor $1520875$ out of $1520875 \sqrt{e}$ .

$p = \frac{1520875 (0)}{1520875 (\sqrt{e})}$

Step 6.3.3.1.2.2

Cancel the common factor.

$p = \frac{1520875 \cdot 0}{1520875 \sqrt{e}}$

Step 6.3.3.1.2.3

Rewrite the expression.

$p = \frac{0}{\sqrt{e}}$

Step 6.3.3.2

Divide $0$ by $\sqrt{e}$ .

$p = 0$

Step 6.4

Set the radicand in $\sqrt{2 p}$ less than $0$ to find where the expression is undefined.

$2 p < 0$

Step 6.5

Divide each term in $2 p < 0$ by $2$ and simplify.

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Step 6.5.1

Divide each term in $2 p < 0$ by $2$ .

$\frac{2 p}{2} < \frac{0}{2}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $2$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{2 p}{2} < \frac{0}{2}$

Step 6.5.2.1.2

Divide $p$ by $1$ .

$p < \frac{0}{2}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $0$ by $2$ .

$p < 0$

Step 6.6

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 $0$ .

$p \leq 0$

$(- \infty, 0]$

$p \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = 512, 0$

Step 8

Evaluate the second derivative at $x = 512$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{\sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$

Step 9

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

No Local Extrema

Step 10