Calculus Examples

$f (x) = x^{\frac{1}{7}} {(x - 1)}^{2}$

Step 1

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

$\frac{d}{d x} [x^{\frac{1}{7}} ((x - 1) (x - 1))]$

Step 2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{1}{7}} (x (x - 1) - 1 (x - 1))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{1}{7}} (x \cdot x + x \cdot - 1 - 1 (x - 1))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{1}{7}} (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)]$

Step 3.1.2

Move $- 1$ to the left of $x$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)]$

Step 3.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} - x - 1 x - 1 \cdot - 1)]$

Step 3.1.4

Rewrite $- 1 x$ as $- x$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} - x - x - 1 \cdot - 1)]$

Step 3.1.5

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} - x - x + 1)]$

Step 3.2

Subtract $x$ from $- x$ .

$\frac{d}{d x} [x^{\frac{1}{7}} (x^{2} - 2 x + 1)]$

Step 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 $f (x) = x^{\frac{1}{7}}$ and $g (x) = x^{2} - 2 x + 1$ .

$x^{\frac{1}{7}} \frac{d}{d x} [x^{2} - 2 x + 1] + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5

Differentiate.

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Step 5.1

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

$x^{\frac{1}{7}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.2

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

$x^{\frac{1}{7}} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.3

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

$x^{\frac{1}{7}} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.4

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

$x^{\frac{1}{7}} (2 x - 2 \cdot 1 + \frac{d}{d x} [1]) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.5

Multiply $- 2$ by $1$ .

$x^{\frac{1}{7}} (2 x - 2 + \frac{d}{d x} [1]) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x^{\frac{1}{7}} (2 x - 2 + 0) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.7

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

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{7}$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{1}{7} - 1})$

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{1}{7} - 1 \cdot \frac{7}{7}})$

Step 7

Combine $- 1$ and $\frac{7}{7}$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{1}{7} + \frac{- 1 \cdot 7}{7}})$

Step 8

Combine the numerators over the common denominator.

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{1 - 1 \cdot 7}{7}})$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $7$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{1 - 7}{7}})$

Step 9.2

Subtract $7$ from $1$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{\frac{- 6}{7}})$

Step 10

Move the negative in front of the fraction.

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) (\frac{1}{7} x^{- \frac{6}{7}})$

Step 11

Combine $\frac{1}{7}$ and $x^{- \frac{6}{7}}$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) \frac{x^{- \frac{6}{7}}}{7}$

Step 12

Move $x^{- \frac{6}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{1}{7}} (2 x - 2) + (x^{2} - 2 x + 1) \frac{1}{7 x^{\frac{6}{7}}}$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$x^{\frac{1}{7}} (2 x) + x^{\frac{1}{7}} \cdot - 2 + (x^{2} - 2 x + 1) \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.2

Apply the distributive property.

$x^{\frac{1}{7}} (2 x) + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3

Combine terms.

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Step 13.3.1

Multiply $x^{\frac{1}{7}}$ by $x$ by adding the exponents.

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Step 13.3.1.1

Move $x$ .

$x \cdot x^{\frac{1}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.1.2

Multiply $x$ by $x^{\frac{1}{7}}$ .

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Step 13.3.1.2.1

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

$x^{1} x^{\frac{1}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.1.2.2

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

$x^{1 + \frac{1}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.1.3

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

$x^{\frac{7}{7} + \frac{1}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.1.4

Combine the numerators over the common denominator.

$x^{\frac{7 + 1}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.1.5

Add $7$ and $1$ .

$x^{\frac{8}{7}} \cdot 2 + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.2

Move $2$ to the left of $x^{\frac{8}{7}}$ .

$2 \cdot x^{\frac{8}{7}} + x^{\frac{1}{7}} \cdot - 2 + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.3

Move $- 2$ to the left of $x^{\frac{1}{7}}$ .

$2 x^{\frac{8}{7}} - 2 \cdot x^{\frac{1}{7}} + x^{2} \frac{1}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.4

Combine $x^{2}$ and $\frac{1}{7 x^{\frac{6}{7}}}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{2}}{7 x^{\frac{6}{7}}} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.5

Move $x^{\frac{6}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{2} x^{- \frac{6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6

Multiply $x^{2}$ by $x^{- \frac{6}{7}}$ by adding the exponents.

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Step 13.3.6.1

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

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{2 - \frac{6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{2 \cdot \frac{7}{7} - \frac{6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6.3

Combine $2$ and $\frac{7}{7}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{2 \cdot 7}{7} - \frac{6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6.4

Combine the numerators over the common denominator.

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{2 \cdot 7 - 6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6.5

Simplify the numerator.

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Step 13.3.6.5.1

Multiply $2$ by $7$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{14 - 6}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.6.5.2

Subtract $6$ from $14$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} - 2 x \frac{1}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.7

Combine $- 2$ and $\frac{1}{7 x^{\frac{6}{7}}}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + x \frac{- 2}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.8

Combine $x$ and $\frac{- 2}{7 x^{\frac{6}{7}}}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{x \cdot - 2}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.9

Move $- 2$ to the left of $x$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 \cdot x}{7 x^{\frac{6}{7}}} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.10

Move $x^{\frac{6}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 x \cdot x^{- \frac{6}{7}}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11

Multiply $x$ by $x^{- \frac{6}{7}}$ by adding the exponents.

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Step 13.3.11.1

Move $x^{- \frac{6}{7}}$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 (x^{- \frac{6}{7}} x)}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11.2

Multiply $x^{- \frac{6}{7}}$ by $x$ .

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Step 13.3.11.2.1

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

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 (x^{- \frac{6}{7}} x^{1})}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11.2.2

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

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 x^{- \frac{6}{7} + 1}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11.3

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

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 x^{- \frac{6}{7} + \frac{7}{7}}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11.4

Combine the numerators over the common denominator.

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 x^{\frac{- 6 + 7}{7}}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.11.5

Add $- 6$ and $7$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} + \frac{- 2 x^{\frac{1}{7}}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.12

Move the negative in front of the fraction.

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + 1 \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.13

Multiply $\frac{1}{7 x^{\frac{6}{7}}}$ by $1$ .

$2 x^{\frac{8}{7}} - 2 x^{\frac{1}{7}} + \frac{x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.14

To write $2 x^{\frac{8}{7}}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- 2 x^{\frac{1}{7}} + 2 x^{\frac{8}{7}} \cdot \frac{7}{7} + \frac{x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.15

Combine $2 x^{\frac{8}{7}}$ and $\frac{7}{7}$ .

$- 2 x^{\frac{1}{7}} + \frac{2 x^{\frac{8}{7}} \cdot 7}{7} + \frac{x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.16

Combine the numerators over the common denominator.

$- 2 x^{\frac{1}{7}} + \frac{2 x^{\frac{8}{7}} \cdot 7 + x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.17

Multiply $7$ by $2$ .

$- 2 x^{\frac{1}{7}} + \frac{14 x^{\frac{8}{7}} + x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.18

Add $14 x^{\frac{8}{7}}$ and $x^{\frac{8}{7}}$ .

$- 2 x^{\frac{1}{7}} + \frac{15 x^{\frac{8}{7}}}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.19

To write $- 2 x^{\frac{1}{7}}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- 2 x^{\frac{1}{7}} \cdot \frac{7}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.20

Combine $- 2 x^{\frac{1}{7}}$ and $\frac{7}{7}$ .

$\frac{- 2 x^{\frac{1}{7}} \cdot 7}{7} - \frac{2 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.21

Combine the numerators over the common denominator.

$\frac{- 2 x^{\frac{1}{7}} \cdot 7 - 2 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.22

Multiply $7$ by $- 2$ .

$\frac{- 14 x^{\frac{1}{7}} - 2 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.23

Subtract $2 x^{\frac{1}{7}}$ from $- 14 x^{\frac{1}{7}}$ .

$\frac{- 16 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.3.24

Move the negative in front of the fraction.

$- \frac{16 x^{\frac{1}{7}}}{7} + \frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}}$

Step 13.4

Reorder terms.

$\frac{15 x^{\frac{8}{7}}}{7} + \frac{1}{7 x^{\frac{6}{7}}} - \frac{16 x^{\frac{1}{7}}}{7}$