Calculus Examples

$x^{\frac{1}{6}}$

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 $n = \frac{1}{6}$ .

$\frac{1}{6} x^{\frac{1}{6} - 1}$

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{1}{6} x^{\frac{1}{6} - 1 \cdot \frac{6}{6}}$

Step 1.3

Combine $- 1$ and $\frac{6}{6}$ .

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

Step 1.4

Combine the numerators over the common denominator.

$\frac{1}{6} x^{\frac{1 - 1 \cdot 6}{6}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $6$ .

$\frac{1}{6} x^{\frac{1 - 6}{6}}$

Step 1.5.2

Subtract $6$ from $1$ .

$\frac{1}{6} x^{\frac{- 5}{6}}$

Step 1.6

Move the negative in front of the fraction.

$\frac{1}{6} x^{- \frac{5}{6}}$

Step 1.7

Simplify.

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Step 1.7.1

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

$\frac{1}{6} \cdot \frac{1}{x^{\frac{5}{6}}}$

Step 1.7.2

Multiply $\frac{1}{6}$ by $\frac{1}{x^{\frac{5}{6}}}$ .

$f' (x) = \frac{1}{6 x^{\frac{5}{6}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6 x^{\frac{5}{6}}}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [\frac{1}{x^{\frac{5}{6}}}]$ .

$\frac{1}{6} \frac{d}{d x} [\frac{1}{x^{\frac{5}{6}}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{5}{6}}}$ as ${(x^{\frac{5}{6}})}^{- 1}$ .

$\frac{1}{6} \frac{d}{d x} [{(x^{\frac{5}{6}})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{5}{6}})}^{- 1}$ .

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Step 2.2.2.1

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

$\frac{1}{6} \frac{d}{d x} [x^{\frac{5}{6} \cdot - 1}]$

Step 2.2.2.2

Multiply $\frac{5}{6} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{5}{6}$ and $- 1$ .

$\frac{1}{6} \frac{d}{d x} [x^{\frac{5 \cdot - 1}{6}}]$

Step 2.2.2.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{6} \frac{d}{d x} [x^{\frac{- 5}{6}}]$

Step 2.2.2.3

Move the negative in front of the fraction.

$\frac{1}{6} \frac{d}{d x} [x^{- \frac{5}{6}}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{5}{6}$ .

$\frac{1}{6} (- \frac{5}{6} x^{- \frac{5}{6} - 1})$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{1}{6} (- \frac{5}{6} x^{- \frac{5}{6} - 1 \cdot \frac{6}{6}})$

Step 2.5

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{1}{6} (- \frac{5}{6} x^{- \frac{5}{6} + \frac{- 1 \cdot 6}{6}})$

Step 2.6

Combine the numerators over the common denominator.

$\frac{1}{6} (- \frac{5}{6} x^{\frac{- 5 - 1 \cdot 6}{6}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $6$ .

$\frac{1}{6} (- \frac{5}{6} x^{\frac{- 5 - 6}{6}})$

Step 2.7.2

Subtract $6$ from $- 5$ .

$\frac{1}{6} (- \frac{5}{6} x^{\frac{- 11}{6}})$

Step 2.8

Move the negative in front of the fraction.

$\frac{1}{6} (- \frac{5}{6} x^{- \frac{11}{6}})$

Step 2.9

Combine $x^{- \frac{11}{6}}$ and $\frac{5}{6}$ .

$\frac{1}{6} (- \frac{x^{- \frac{11}{6}} \cdot 5}{6})$

Step 2.10

Multiply $\frac{1}{6}$ by $\frac{x^{- \frac{11}{6}} \cdot 5}{6}$ .

$- \frac{x^{- \frac{11}{6}} \cdot 5}{6 \cdot 6}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Multiply $6$ by $6$ .

$- \frac{x^{- \frac{11}{6}} \cdot 5}{36}$

Step 2.11.2

Move $5$ to the left of $x^{- \frac{11}{6}}$ .

$- \frac{5 \cdot x^{- \frac{11}{6}}}{36}$

Step 2.11.3

Move $x^{- \frac{11}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{5}{36 x^{\frac{11}{6}}}$