Calculus Examples

$f (x) = x^{\frac{5}{7}}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{7}$ .

$\frac{5}{7} x^{\frac{5}{7} - 1}$

Step 1.1.2

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

$\frac{5}{7} x^{\frac{5}{7} - 1 \cdot \frac{7}{7}}$

Step 1.1.3

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

$\frac{5}{7} x^{\frac{5}{7} + \frac{- 1 \cdot 7}{7}}$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{5}{7} x^{\frac{5 - 1 \cdot 7}{7}}$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $7$ .

$\frac{5}{7} x^{\frac{5 - 7}{7}}$

Step 1.1.5.2

Subtract $7$ from $5$ .

$\frac{5}{7} x^{\frac{- 2}{7}}$

Step 1.1.6

Move the negative in front of the fraction.

$\frac{5}{7} x^{- \frac{2}{7}}$

Step 1.1.7

Simplify.

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Step 1.1.7.1

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

$\frac{5}{7} \cdot \frac{1}{x^{\frac{2}{7}}}$

Step 1.1.7.2

Multiply $\frac{5}{7}$ by $\frac{1}{x^{\frac{2}{7}}}$ .

$f' (x) = \frac{5}{7 x^{\frac{2}{7}}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $\frac{5}{7}$ is constant with respect to $x$ , the derivative of $\frac{5}{7 x^{\frac{2}{7}}}$ with respect to $x$ is $\frac{5}{7} \frac{d}{d x} [\frac{1}{x^{\frac{2}{7}}}]$ .

$\frac{5}{7} \frac{d}{d x} [\frac{1}{x^{\frac{2}{7}}}]$

Step 1.2.2

Apply basic rules of exponents.

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Step 1.2.2.1

Rewrite $\frac{1}{x^{\frac{2}{7}}}$ as ${(x^{\frac{2}{7}})}^{- 1}$ .

$\frac{5}{7} \frac{d}{d x} [{(x^{\frac{2}{7}})}^{- 1}]$

Step 1.2.2.2

Multiply the exponents in ${(x^{\frac{2}{7}})}^{- 1}$ .

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Step 1.2.2.2.1

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

$\frac{5}{7} \frac{d}{d x} [x^{\frac{2}{7} \cdot - 1}]$

Step 1.2.2.2.2

Multiply $\frac{2}{7} \cdot - 1$ .

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Step 1.2.2.2.2.1

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

$\frac{5}{7} \frac{d}{d x} [x^{\frac{2 \cdot - 1}{7}}]$

Step 1.2.2.2.2.2

Multiply $2$ by $- 1$ .

$\frac{5}{7} \frac{d}{d x} [x^{\frac{- 2}{7}}]$

Step 1.2.2.2.3

Move the negative in front of the fraction.

$\frac{5}{7} \frac{d}{d x} [x^{- \frac{2}{7}}]$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{2}{7}$ .

$\frac{5}{7} (- \frac{2}{7} x^{- \frac{2}{7} - 1})$

Step 1.2.4

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

$\frac{5}{7} (- \frac{2}{7} x^{- \frac{2}{7} - 1 \cdot \frac{7}{7}})$

Step 1.2.5

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

$\frac{5}{7} (- \frac{2}{7} x^{- \frac{2}{7} + \frac{- 1 \cdot 7}{7}})$

Step 1.2.6

Combine the numerators over the common denominator.

$\frac{5}{7} (- \frac{2}{7} x^{\frac{- 2 - 1 \cdot 7}{7}})$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $7$ .

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

Step 1.2.7.2

Subtract $7$ from $- 2$ .

$\frac{5}{7} (- \frac{2}{7} x^{\frac{- 9}{7}})$

Step 1.2.8

Move the negative in front of the fraction.

$\frac{5}{7} (- \frac{2}{7} x^{- \frac{9}{7}})$

Step 1.2.9

Combine $x^{- \frac{9}{7}}$ and $\frac{2}{7}$ .

$\frac{5}{7} (- \frac{x^{- \frac{9}{7}} \cdot 2}{7})$

Step 1.2.10

Multiply $\frac{5}{7}$ by $\frac{x^{- \frac{9}{7}} \cdot 2}{7}$ .

$- \frac{5 (x^{- \frac{9}{7}} \cdot 2)}{7 \cdot 7}$

Step 1.2.11

Multiply.

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Step 1.2.11.1

Multiply $2$ by $5$ .

$- \frac{10 x^{- \frac{9}{7}}}{7 \cdot 7}$

Step 1.2.11.2

Multiply $7$ by $7$ .

$- \frac{10 x^{- \frac{9}{7}}}{49}$

Step 1.2.11.3

Move $x^{- \frac{9}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{10}{49 x^{\frac{9}{7}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{10}{49 x^{\frac{9}{7}}}$ .

$- \frac{10}{49 x^{\frac{9}{7}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{10}{49 x^{\frac{9}{7}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{10}{49 x^{\frac{9}{7}}} = 0$

Step 2.2

Set the numerator equal to zero.

$10 = 0$

Step 2.3

Since $10 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points