Calculus Examples

$x^{\frac{22}{7}}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{22}{7}$ .

$\frac{22}{7} x^{\frac{22}{7} - 1}$

Step 1.2

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

$\frac{22}{7} x^{\frac{22}{7} - 1 \cdot \frac{7}{7}}$

Step 1.3

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

$\frac{22}{7} x^{\frac{22}{7} + \frac{- 1 \cdot 7}{7}}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{22}{7} x^{\frac{22 - 1 \cdot 7}{7}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $7$ .

$\frac{22}{7} x^{\frac{22 - 7}{7}}$

Step 1.5.2

Subtract $7$ from $22$ .

$\frac{22}{7} x^{\frac{15}{7}}$

Step 1.6

Combine $\frac{22}{7}$ and $x^{\frac{15}{7}}$ .

$f' (x) = \frac{22 x^{\frac{15}{7}}}{7}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{22}{7}$ is constant with respect to $x$ , the derivative of $\frac{22 x^{\frac{15}{7}}}{7}$ with respect to $x$ is $\frac{22}{7} \frac{d}{d x} [x^{\frac{15}{7}}]$ .

$\frac{22}{7} \frac{d}{d x} [x^{\frac{15}{7}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{15}{7}$ .

$\frac{22}{7} (\frac{15}{7} x^{\frac{15}{7} - 1})$

Step 2.3

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

$\frac{22}{7} (\frac{15}{7} x^{\frac{15}{7} - 1 \cdot \frac{7}{7}})$

Step 2.4

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

$\frac{22}{7} (\frac{15}{7} x^{\frac{15}{7} + \frac{- 1 \cdot 7}{7}})$

Step 2.5

Combine the numerators over the common denominator.

$\frac{22}{7} (\frac{15}{7} x^{\frac{15 - 1 \cdot 7}{7}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $7$ .

$\frac{22}{7} (\frac{15}{7} x^{\frac{15 - 7}{7}})$

Step 2.6.2

Subtract $7$ from $15$ .

$\frac{22}{7} (\frac{15}{7} x^{\frac{8}{7}})$

Step 2.7

Combine $\frac{15}{7}$ and $x^{\frac{8}{7}}$ .

$\frac{22}{7} \cdot \frac{15 x^{\frac{8}{7}}}{7}$

Step 2.8

Multiply $\frac{22}{7}$ by $\frac{15 x^{\frac{8}{7}}}{7}$ .

$\frac{22 (15 x^{\frac{8}{7}})}{7 \cdot 7}$

Step 2.9

Multiply.

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Step 2.9.1

Multiply $15$ by $22$ .

$\frac{330 x^{\frac{8}{7}}}{7 \cdot 7}$

Step 2.9.2

Multiply $7$ by $7$ .

$f'' (x) = \frac{330 x^{\frac{8}{7}}}{49}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{330}{49}$ is constant with respect to $x$ , the derivative of $\frac{330 x^{\frac{8}{7}}}{49}$ with respect to $x$ is $\frac{330}{49} \frac{d}{d x} [x^{\frac{8}{7}}]$ .

$\frac{330}{49} \frac{d}{d x} [x^{\frac{8}{7}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{7}$ .

$\frac{330}{49} (\frac{8}{7} x^{\frac{8}{7} - 1})$

Step 3.3

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

$\frac{330}{49} (\frac{8}{7} x^{\frac{8}{7} - 1 \cdot \frac{7}{7}})$

Step 3.4

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

$\frac{330}{49} (\frac{8}{7} x^{\frac{8}{7} + \frac{- 1 \cdot 7}{7}})$

Step 3.5

Combine the numerators over the common denominator.

$\frac{330}{49} (\frac{8}{7} x^{\frac{8 - 1 \cdot 7}{7}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $7$ .

$\frac{330}{49} (\frac{8}{7} x^{\frac{8 - 7}{7}})$

Step 3.6.2

Subtract $7$ from $8$ .

$\frac{330}{49} (\frac{8}{7} x^{\frac{1}{7}})$

Step 3.7

Combine $\frac{8}{7}$ and $x^{\frac{1}{7}}$ .

$\frac{330}{49} \cdot \frac{8 x^{\frac{1}{7}}}{7}$

Step 3.8

Multiply $\frac{330}{49}$ by $\frac{8 x^{\frac{1}{7}}}{7}$ .

$\frac{330 (8 x^{\frac{1}{7}})}{49 \cdot 7}$

Step 3.9

Multiply.

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Step 3.9.1

Multiply $8$ by $330$ .

$\frac{2640 x^{\frac{1}{7}}}{49 \cdot 7}$

Step 3.9.2

Multiply $49$ by $7$ .

$f''' (x) = \frac{2640 x^{\frac{1}{7}}}{343}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{2640}{343}$ is constant with respect to $x$ , the derivative of $\frac{2640 x^{\frac{1}{7}}}{343}$ with respect to $x$ is $\frac{2640}{343} \frac{d}{d x} [x^{\frac{1}{7}}]$ .

$\frac{2640}{343} \frac{d}{d x} [x^{\frac{1}{7}}]$

Step 4.2

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

$\frac{2640}{343} (\frac{1}{7} x^{\frac{1}{7} - 1})$

Step 4.3

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

$\frac{2640}{343} (\frac{1}{7} x^{\frac{1}{7} - 1 \cdot \frac{7}{7}})$

Step 4.4

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

$\frac{2640}{343} (\frac{1}{7} x^{\frac{1}{7} + \frac{- 1 \cdot 7}{7}})$

Step 4.5

Combine the numerators over the common denominator.

$\frac{2640}{343} (\frac{1}{7} x^{\frac{1 - 1 \cdot 7}{7}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $7$ .

$\frac{2640}{343} (\frac{1}{7} x^{\frac{1 - 7}{7}})$

Step 4.6.2

Subtract $7$ from $1$ .

$\frac{2640}{343} (\frac{1}{7} x^{\frac{- 6}{7}})$

Step 4.7

Move the negative in front of the fraction.

$\frac{2640}{343} (\frac{1}{7} x^{- \frac{6}{7}})$

Step 4.8

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

$\frac{2640}{343} \cdot \frac{x^{- \frac{6}{7}}}{7}$

Step 4.9

Multiply $\frac{2640}{343}$ by $\frac{x^{- \frac{6}{7}}}{7}$ .

$\frac{2640 x^{- \frac{6}{7}}}{343 \cdot 7}$

Step 4.10

Multiply.

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Step 4.10.1

Multiply $343$ by $7$ .

$\frac{2640 x^{- \frac{6}{7}}}{2401}$

Step 4.10.2

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

$f^{4} (x) = \frac{2640}{2401 x^{\frac{6}{7}}}$