Calculus Examples

$x^{\frac{4}{5}} {(x - 5)}^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$\frac{d}{d x} [x^{\frac{4}{5}} ((x - 5) (x - 5))]$

Step 1.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{4}{5}} (x (x - 5) - 5 (x - 5))]$

Step 1.1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{4}{5}} (x \cdot x + x \cdot - 5 - 5 (x - 5))]$

Step 1.1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{4}{5}} (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{\frac{4}{5}} (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 1.1.3.1.2

Move $- 5$ to the left of $x$ .

$\frac{d}{d x} [x^{\frac{4}{5}} (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)]$

Step 1.1.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [x^{\frac{4}{5}} (x^{2} - 5 x - 5 x + 25)]$

Step 1.1.3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{d}{d x} [x^{\frac{4}{5}} (x^{2} - 10 x + 25)]$

Step 1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{4}{5}}$ and $g (x) = x^{2} - 10 x + 25$ .

$x^{\frac{4}{5}} \frac{d}{d x} [x^{2} - 10 x + 25] + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

By the Sum Rule, the derivative of $x^{2} - 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]$ .

$x^{\frac{4}{5}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.2

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

$x^{\frac{4}{5}} (2 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.3

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

$x^{\frac{4}{5}} (2 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.4

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

$x^{\frac{4}{5}} (2 x - 10 \cdot 1 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.5

Multiply $- 10$ by $1$ .

$x^{\frac{4}{5}} (2 x - 10 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$x^{\frac{4}{5}} (2 x - 10 + 0) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.7

Add $2 x - 10$ and $0$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{4}{5} - 1})$

Step 1.1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 1.1.7

Combine $- 1$ and $\frac{5}{5}$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 1.1.8

Combine the numerators over the common denominator.

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}})$

Step 1.1.9

Simplify the numerator.

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Step 1.1.9.1

Multiply $- 1$ by $5$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{4 - 5}{5}})$

Step 1.1.9.2

Subtract $5$ from $4$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{\frac{- 1}{5}})$

Step 1.1.10

Move the negative in front of the fraction.

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{4}{5} x^{- \frac{1}{5}})$

Step 1.1.11

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{4 x^{- \frac{1}{5}}}{5}$

Step 1.1.12

Move $x^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{4}{5}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13

Simplify.

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Step 1.1.13.1

Apply the distributive property.

$x^{\frac{4}{5}} (2 x) + x^{\frac{4}{5}} \cdot - 10 + (x^{2} - 10 x + 25) \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.2

Apply the distributive property.

$x^{\frac{4}{5}} (2 x) + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3

Combine terms.

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Step 1.1.13.3.1

Multiply $x^{\frac{4}{5}}$ by $x$ by adding the exponents.

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Step 1.1.13.3.1.1

Move $x$ .

$x \cdot x^{\frac{4}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.1.2

Multiply $x$ by $x^{\frac{4}{5}}$ .

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Step 1.1.13.3.1.2.1

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

$x^{1} x^{\frac{4}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.1.2.2

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

$x^{1 + \frac{4}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.1.3

Write $1$ as a fraction with a common denominator.

$x^{\frac{5}{5} + \frac{4}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.1.4

Combine the numerators over the common denominator.

$x^{\frac{5 + 4}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.1.5

Add $5$ and $4$ .

$x^{\frac{9}{5}} \cdot 2 + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.2

Move $2$ to the left of $x^{\frac{9}{5}}$ .

$2 \cdot x^{\frac{9}{5}} + x^{\frac{4}{5}} \cdot - 10 + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.3

Move $- 10$ to the left of $x^{\frac{4}{5}}$ .

$2 x^{\frac{9}{5}} - 10 \cdot x^{\frac{4}{5}} + x^{2} \frac{4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.4

Combine $x^{2}$ and $\frac{4}{5 x^{\frac{1}{5}}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{x^{2} \cdot 4}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.5

Move $4$ to the left of $x^{2}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 \cdot x^{2}}{5 x^{\frac{1}{5}}} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.6

Move $x^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{2} x^{- \frac{1}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7

Multiply $x^{2}$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 1.1.13.3.7.1

Move $x^{- \frac{1}{5}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 (x^{- \frac{1}{5}} x^{2})}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.2

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

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{- \frac{1}{5} + 2}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{- \frac{1}{5} + 2 \cdot \frac{5}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.4

Combine $2$ and $\frac{5}{5}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{- \frac{1}{5} + \frac{2 \cdot 5}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.5

Combine the numerators over the common denominator.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{- 1 + 2 \cdot 5}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.6

Simplify the numerator.

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Step 1.1.13.3.7.6.1

Multiply $2$ by $5$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{- 1 + 10}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.7.6.2

Add $- 1$ and $10$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 10 x \frac{4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.8

Combine $- 10$ and $\frac{4}{5 x^{\frac{1}{5}}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + x \frac{- 10 \cdot 4}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.9

Multiply $- 10$ by $4$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + x \frac{- 40}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.10

Combine $x$ and $\frac{- 40}{5 x^{\frac{1}{5}}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{x \cdot - 40}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.11

Move $- 40$ to the left of $x$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 \cdot x}{5 x^{\frac{1}{5}}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.12

Move $x^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 x \cdot x^{- \frac{1}{5}}}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13

Multiply $x$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 1.1.13.3.13.1

Move $x^{- \frac{1}{5}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 (x^{- \frac{1}{5}} x)}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13.2

Multiply $x^{- \frac{1}{5}}$ by $x$ .

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Step 1.1.13.3.13.2.1

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

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 (x^{- \frac{1}{5}} x^{1})}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13.2.2

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

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 x^{- \frac{1}{5} + 1}}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13.3

Write $1$ as a fraction with a common denominator.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 x^{- \frac{1}{5} + \frac{5}{5}}}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13.4

Combine the numerators over the common denominator.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 x^{\frac{- 1 + 5}{5}}}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.13.5

Add $- 1$ and $5$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 40 x^{\frac{4}{5}}}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.14

Factor $5$ out of $- 40 x^{\frac{4}{5}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{5 (- 8 x^{\frac{4}{5}})}{5} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.15

Cancel the common factors.

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Step 1.1.13.3.15.1

Factor $5$ out of $5$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{5 (- 8 x^{\frac{4}{5}})}{5 (1)} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.15.2

Cancel the common factor.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{5 (- 8 x^{\frac{4}{5}})}{5 \cdot 1} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.15.3

Rewrite the expression.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} + \frac{- 8 x^{\frac{4}{5}}}{1} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.15.4

Divide $- 8 x^{\frac{4}{5}}$ by $1$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + 25 \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.16

Combine $25$ and $\frac{4}{5 x^{\frac{1}{5}}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{25 \cdot 4}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.17

Multiply $25$ by $4$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{100}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.18

Factor $5$ out of $100$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{5 \cdot 20}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.19

Cancel the common factors.

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Step 1.1.13.3.19.1

Factor $5$ out of $5 x^{\frac{1}{5}}$ .

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{5 \cdot 20}{5 (x^{\frac{1}{5}})}$

Step 1.1.13.3.19.2

Cancel the common factor.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{5 \cdot 20}{5 x^{\frac{1}{5}}}$

Step 1.1.13.3.19.3

Rewrite the expression.

$2 x^{\frac{9}{5}} - 10 x^{\frac{4}{5}} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.20

To write $2 x^{\frac{9}{5}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 10 x^{\frac{4}{5}} + 2 x^{\frac{9}{5}} \cdot \frac{5}{5} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.21

Combine $2 x^{\frac{9}{5}}$ and $\frac{5}{5}$ .

$- 10 x^{\frac{4}{5}} + \frac{2 x^{\frac{9}{5}} \cdot 5}{5} + \frac{4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.22

Combine the numerators over the common denominator.

$- 10 x^{\frac{4}{5}} + \frac{2 x^{\frac{9}{5}} \cdot 5 + 4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.23

Multiply $5$ by $2$ .

$- 10 x^{\frac{4}{5}} + \frac{10 x^{\frac{9}{5}} + 4 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.24

Add $10 x^{\frac{9}{5}}$ and $4 x^{\frac{9}{5}}$ .

$- 10 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} - 8 x^{\frac{4}{5}} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.1.13.3.25

Subtract $8 x^{\frac{4}{5}}$ from $- 10 x^{\frac{4}{5}}$ .

$f' (x) = - 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}}$ .

$- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$1, 5, x^{\frac{1}{5}}, 1$

Step 2.2.2

Since $1, 5, x^{\frac{1}{5}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 5, 1, 1$ then find LCM for the variable part $x^{\frac{1}{5}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.5

Since $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

Step 2.2.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.7

The LCM of $1, 5, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$5$

Step 2.2.8

The LCM of $x^{\frac{1}{5}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{1}{5}}$

Step 2.2.9

The LCM for $1, 5, x^{\frac{1}{5}}, 1$ is the numeric part $5$ multiplied by the variable part.

$5 x^{\frac{1}{5}}$

Step 2.3

Multiply each term in $- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}} = 0$ by $5 x^{\frac{1}{5}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $- 18 x^{\frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}} = 0$ by $5 x^{\frac{1}{5}}$ .

$- 18 x^{\frac{4}{5}} (5 x^{\frac{1}{5}}) + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

$- 18 \cdot 5 x^{\frac{4}{5}} x^{\frac{1}{5}} + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.2

Multiply $x^{\frac{4}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 2.3.2.1.2.1

Move $x^{\frac{1}{5}}$ .

$- 18 \cdot 5 (x^{\frac{1}{5}} x^{\frac{4}{5}}) + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.2.2

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

$- 18 \cdot 5 x^{\frac{1}{5} + \frac{4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.2.3

Combine the numerators over the common denominator.

$- 18 \cdot 5 x^{\frac{1 + 4}{5}} + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.2.4

Add $1$ and $4$ .

$- 18 \cdot 5 x^{\frac{5}{5}} + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.2.5

Divide $5$ by $5$ .

$- 18 \cdot 5 x^{1} + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.3

Simplify $- 18 \cdot 5 x^{1}$ .

$- 18 \cdot 5 x + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.4

Multiply $- 18$ by $5$ .

$- 90 x + \frac{14 x^{\frac{9}{5}}}{5} (5 x^{\frac{1}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.5

Rewrite using the commutative property of multiplication.

$- 90 x + 5 \frac{14 x^{\frac{9}{5}}}{5} x^{\frac{1}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.6

Cancel the common factor of $5$ .

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Step 2.3.2.1.6.1

Cancel the common factor.

$- 90 x + 5 \frac{14 x^{\frac{9}{5}}}{5} x^{\frac{1}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.6.2

Rewrite the expression.

$- 90 x + 14 x^{\frac{9}{5}} x^{\frac{1}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.7

Multiply $x^{\frac{9}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 2.3.2.1.7.1

Move $x^{\frac{1}{5}}$ .

$- 90 x + 14 (x^{\frac{1}{5}} x^{\frac{9}{5}}) + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.7.2

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

$- 90 x + 14 x^{\frac{1}{5} + \frac{9}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.7.3

Combine the numerators over the common denominator.

$- 90 x + 14 x^{\frac{1 + 9}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.7.4

Add $1$ and $9$ .

$- 90 x + 14 x^{\frac{10}{5}} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.7.5

Divide $10$ by $5$ .

$- 90 x + 14 x^{2} + \frac{20}{x^{\frac{1}{5}}} (5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.8

Rewrite using the commutative property of multiplication.

$- 90 x + 14 x^{2} + 5 \frac{20}{x^{\frac{1}{5}}} x^{\frac{1}{5}} = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.9

Multiply $5 \frac{20}{x^{\frac{1}{5}}}$ .

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Step 2.3.2.1.9.1

Combine $5$ and $\frac{20}{x^{\frac{1}{5}}}$ .

$- 90 x + 14 x^{2} + \frac{5 \cdot 20}{x^{\frac{1}{5}}} x^{\frac{1}{5}} = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.9.2

Multiply $5$ by $20$ .

$- 90 x + 14 x^{2} + \frac{100}{x^{\frac{1}{5}}} x^{\frac{1}{5}} = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.10

Cancel the common factor of $x^{\frac{1}{5}}$ .

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Step 2.3.2.1.10.1

Cancel the common factor.

$- 90 x + 14 x^{2} + \frac{100}{x^{\frac{1}{5}}} x^{\frac{1}{5}} = 0 (5 x^{\frac{1}{5}})$

Step 2.3.2.1.10.2

Rewrite the expression.

$- 90 x + 14 x^{2} + 100 = 0 (5 x^{\frac{1}{5}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (5 x^{\frac{1}{5}})$ .

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Step 2.3.3.1.1

Multiply $5$ by $0$ .

$- 90 x + 14 x^{2} + 100 = 0 x^{\frac{1}{5}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{1}{5}}$ .

$- 90 x + 14 x^{2} + 100 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Factor the left side of the equation.

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Step 2.4.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$- 90 u + 14 u^{2} + 100 = 0$

Step 2.4.1.2

Factor $2$ out of $- 90 u + 14 u^{2} + 100$ .

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Step 2.4.1.2.1

Factor $2$ out of $- 90 u$ .

$2 (- 45 u) + 14 u^{2} + 100 = 0$

Step 2.4.1.2.2

Factor $2$ out of $14 u^{2}$ .

$2 (- 45 u) + 2 (7 u^{2}) + 100 = 0$

Step 2.4.1.2.3

Factor $2$ out of $100$ .

$2 (- 45 u) + 2 (7 u^{2}) + 2 (50) = 0$

Step 2.4.1.2.4

Factor $2$ out of $2 (- 45 u) + 2 (7 u^{2})$ .

$2 (- 45 u + 7 u^{2}) + 2 (50) = 0$

Step 2.4.1.2.5

Factor $2$ out of $2 (- 45 u + 7 u^{2}) + 2 (50)$ .

$2 (- 45 u + 7 u^{2} + 50) = 0$

Step 2.4.1.3

Factor.

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Step 2.4.1.3.1

Factor by grouping.

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Step 2.4.1.3.1.1

Reorder terms.

$2 (7 u^{2} - 45 u + 50) = 0$

Step 2.4.1.3.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot 50 = 350$ and whose sum is $b = - 45$ .

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Step 2.4.1.3.1.2.1

Factor $- 45$ out of $- 45 u$ .

$2 (7 u^{2} - 45 u + 50) = 0$

Step 2.4.1.3.1.2.2

Rewrite $- 45$ as $- 10$ plus $- 35$

$2 (7 u^{2} + (- 10 - 35) u + 50) = 0$

Step 2.4.1.3.1.2.3

Apply the distributive property.

$2 (7 u^{2} - 10 u - 35 u + 50) = 0$

Step 2.4.1.3.1.3

Factor out the greatest common factor from each group.

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Step 2.4.1.3.1.3.1

Group the first two terms and the last two terms.

$2 ((7 u^{2} - 10 u) - 35 u + 50) = 0$

Step 2.4.1.3.1.3.2

Factor out the greatest common factor (GCF) from each group.

$2 (u (7 u - 10) - 5 (7 u - 10)) = 0$

Step 2.4.1.3.1.4

Factor the polynomial by factoring out the greatest common factor, $7 u - 10$ .

$2 ((7 u - 10) (u - 5)) = 0$

Step 2.4.1.3.2

Remove unnecessary parentheses.

$2 (7 u - 10) (u - 5) = 0$

Step 2.4.1.4

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

$2 (7 x - 10) (x - 5) = 0$

Step 2.4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$7 x - 10 = 0$

$x - 5 = 0$

Step 2.4.3

Set $7 x - 10$ equal to $0$ and solve for $x$ .

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Step 2.4.3.1

Set $7 x - 10$ equal to $0$ .

$7 x - 10 = 0$

Step 2.4.3.2

Solve $7 x - 10 = 0$ for $x$ .

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Step 2.4.3.2.1

Add $10$ to both sides of the equation.

$7 x = 10$

Step 2.4.3.2.2

Divide each term in $7 x = 10$ by $7$ and simplify.

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Step 2.4.3.2.2.1

Divide each term in $7 x = 10$ by $7$ .

$\frac{7 x}{7} = \frac{10}{7}$

Step 2.4.3.2.2.2

Simplify the left side.

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Step 2.4.3.2.2.2.1

Cancel the common factor of $7$ .

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Step 2.4.3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{10}{7}$

Step 2.4.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{7}$

Step 2.4.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.4.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.4.5

The final solution is all the values that make $2 (7 x - 10) (x - 5) = 0$ true.

$x = \frac{10}{7}, 5$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$- 18 \sqrt[5]{x^{4}} + \frac{14 x^{\frac{9}{5}}}{5} + \frac{20}{x^{\frac{1}{5}}}$

Step 3.1.2

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

$- 18 \sqrt[5]{x^{4}} + \frac{14 \sqrt[5]{x^{9}}}{5} + \frac{20}{x^{\frac{1}{5}}}$

Step 3.1.3

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

$- 18 \sqrt[5]{x^{4}} + \frac{14 \sqrt[5]{x^{9}}}{5} + \frac{20}{\sqrt[5]{x^{1}}}$

Step 3.1.4

Anything raised to $1$ is the base itself.

$- 18 \sqrt[5]{x^{4}} + \frac{14 \sqrt[5]{x^{9}}}{5} + \frac{20}{\sqrt[5]{x}}$

Step 3.2

Set the denominator in $\frac{20}{\sqrt[5]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[5]{x} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{x}}^{5} = 0^{5}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

${(x^{\frac{1}{5}})}^{5} = 0^{5}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.2.1.1.1

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

$x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{5}$

Step 3.3.2.2.1.2

Simplify.

$x = 0^{5}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 4

Evaluate $x^{\frac{4}{5}} {(x - 5)}^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{10}{7}$ .

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Step 4.1.1

Substitute $\frac{10}{7}$ for $x$ .

${(\frac{10}{7})}^{\frac{4}{5}} {((\frac{10}{7}) - 5)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Apply the product rule to $\frac{10}{7}$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {((\frac{10}{7}) - 5)}^{2}$

Step 4.1.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(\frac{10}{7} - 5 \cdot \frac{7}{7})}^{2}$

Step 4.1.2.3

Combine $- 5$ and $\frac{7}{7}$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(\frac{10}{7} + \frac{- 5 \cdot 7}{7})}^{2}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(\frac{10 - 5 \cdot 7}{7})}^{2}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $- 5$ by $7$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(\frac{10 - 35}{7})}^{2}$

Step 4.1.2.5.2

Subtract $35$ from $10$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(\frac{- 25}{7})}^{2}$

Step 4.1.2.6

Move the negative in front of the fraction.

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} {(- \frac{25}{7})}^{2}$

Step 4.1.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.7.1

Apply the product rule to $- \frac{25}{7}$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} ({(- 1)}^{2} {(\frac{25}{7})}^{2})$

Step 4.1.2.7.2

Apply the product rule to $\frac{25}{7}$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} ({(- 1)}^{2} \frac{25^{2}}{7^{2}})$

Step 4.1.2.8

Simplify the expression.

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Step 4.1.2.8.1

Raise $- 1$ to the power of $2$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} (1 \frac{25^{2}}{7^{2}})$

Step 4.1.2.8.2

Multiply $\frac{25^{2}}{7^{2}}$ by $1$ .

$\frac{10^{\frac{4}{5}}}{7^{\frac{4}{5}}} \cdot \frac{25^{2}}{7^{2}}$

Step 4.1.2.9

Combine.

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4}{5}} \cdot 7^{2}}$

Step 4.1.2.10

Multiply $7^{\frac{4}{5}}$ by $7^{2}$ by adding the exponents.

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Step 4.1.2.10.1

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

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4}{5} + 2}}$

Step 4.1.2.10.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4}{5} + 2 \cdot \frac{5}{5}}}$

Step 4.1.2.10.3

Combine $2$ and $\frac{5}{5}$ .

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4}{5} + \frac{2 \cdot 5}{5}}}$

Step 4.1.2.10.4

Combine the numerators over the common denominator.

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4 + 2 \cdot 5}{5}}}$

Step 4.1.2.10.5

Simplify the numerator.

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Step 4.1.2.10.5.1

Multiply $2$ by $5$ .

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{4 + 10}{5}}}$

Step 4.1.2.10.5.2

Add $4$ and $10$ .

$\frac{10^{\frac{4}{5}} \cdot 25^{2}}{7^{\frac{14}{5}}}$

Step 4.1.2.11

Raise $25$ to the power of $2$ .

$\frac{10^{\frac{4}{5}} \cdot 625}{7^{\frac{14}{5}}}$

Step 4.1.2.12

Move $625$ to the left of $10^{\frac{4}{5}}$ .

$\frac{625 \cdot 10^{\frac{4}{5}}}{7^{\frac{14}{5}}}$

Step 4.2

Evaluate at $x = 5$ .

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Step 4.2.1

Substitute $5$ for $x$ .

${(5)}^{\frac{4}{5}} {((5) - 5)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $5$ from $5$ .

$5^{\frac{4}{5}} \cdot 0^{2}$

Step 4.2.2.2

Raising $0$ to any positive power yields $0$ .

$5^{\frac{4}{5}} \cdot 0$

Step 4.2.2.3

Multiply $5^{\frac{4}{5}}$ by $0$ .

$0$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

${(0)}^{\frac{4}{5}} {((0) - 5)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Rewrite $0$ as $0^{5}$ .

${(0^{5})}^{\frac{4}{5}} {((0) - 5)}^{2}$

Step 4.3.2.1.2

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

$0^{5 (\frac{4}{5})} {((0) - 5)}^{2}$

Step 4.3.2.2

Cancel the common factor of $5$ .

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Step 4.3.2.2.1

Cancel the common factor.

$0^{5 (\frac{4}{5})} {((0) - 5)}^{2}$

Step 4.3.2.2.2

Rewrite the expression.

$0^{4} {((0) - 5)}^{2}$

Step 4.3.2.3

Simplify the expression.

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Step 4.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$0 {((0) - 5)}^{2}$

Step 4.3.2.3.2

Subtract $5$ from $0$ .

$0 {(- 5)}^{2}$

Step 4.3.2.3.3

Raise $- 5$ to the power of $2$ .

$0 \cdot 25$

Step 4.3.2.3.4

Multiply $0$ by $25$ .

$0$

Step 4.4

List all of the points.

$(\frac{10}{7}, \frac{625 \cdot 10^{\frac{4}{5}}}{7^{\frac{14}{5}}}), (5, 0), (0, 0)$

Step 5