Calculus Examples

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

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.

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Step 1.1.1.1

By the Sum Rule, the derivative of $- 5 - x^{\frac{4}{5}}$ with respect to $x$ is $\frac{d}{d x} [- 5] + \frac{d}{d x} [- x^{\frac{4}{5}}]$ .

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

Step 1.1.1.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- x^{\frac{4}{5}}]$ .

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{4}{5}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$0 - \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.1.2.2

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

$0 - (\frac{4}{5} x^{\frac{4}{5} - 1})$

Step 1.1.2.3

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

$0 - (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 1.1.2.4

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

$0 - (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 1.1.2.5

Combine the numerators over the common denominator.

$0 - (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}})$

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $5$ .

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

Step 1.1.2.6.2

Subtract $5$ from $4$ .

$0 - (\frac{4}{5} x^{\frac{- 1}{5}})$

Step 1.1.2.7

Move the negative in front of the fraction.

$0 - (\frac{4}{5} x^{- \frac{1}{5}})$

Step 1.1.2.8

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

$0 - \frac{4 x^{- \frac{1}{5}}}{5}$

Step 1.1.2.9

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

$0 - \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.1.3

Subtract $\frac{4}{5 x^{\frac{1}{5}}}$ from $0$ .

$f' (x) = - \frac{4}{5 x^{\frac{1}{5}}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $- \frac{4}{5}$ is constant with respect to $x$ , the derivative of $- \frac{4}{5 x^{\frac{1}{5}}}$ with respect to $x$ is $- \frac{4}{5} \frac{d}{d x} [\frac{1}{x^{\frac{1}{5}}}]$ .

$- \frac{4}{5} \frac{d}{d x} [\frac{1}{x^{\frac{1}{5}}}]$

Step 1.2.2

Apply basic rules of exponents.

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Step 1.2.2.1

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

$- \frac{4}{5} \frac{d}{d x} [{(x^{\frac{1}{5}})}^{- 1}]$

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$- \frac{4}{5} \frac{d}{d x} [x^{\frac{1}{5} \cdot - 1}]$

Step 1.2.2.2.2

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

$- \frac{4}{5} \frac{d}{d x} [x^{\frac{- 1}{5}}]$

Step 1.2.2.2.3

Move the negative in front of the fraction.

$- \frac{4}{5} \frac{d}{d x} [x^{- \frac{1}{5}}]$

Step 1.2.3

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

$- \frac{4}{5} (- \frac{1}{5} x^{- \frac{1}{5} - 1})$

Step 1.2.4

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

$- \frac{4}{5} (- \frac{1}{5} x^{- \frac{1}{5} - 1 \cdot \frac{5}{5}})$

Step 1.2.5

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

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

Step 1.2.6

Combine the numerators over the common denominator.

$- \frac{4}{5} (- \frac{1}{5} x^{\frac{- 1 - 1 \cdot 5}{5}})$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $5$ .

$- \frac{4}{5} (- \frac{1}{5} x^{\frac{- 1 - 5}{5}})$

Step 1.2.7.2

Subtract $5$ from $- 1$ .

$- \frac{4}{5} (- \frac{1}{5} x^{\frac{- 6}{5}})$

Step 1.2.8

Move the negative in front of the fraction.

$- \frac{4}{5} (- \frac{1}{5} x^{- \frac{6}{5}})$

Step 1.2.9

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

$- \frac{4}{5} (- \frac{x^{- \frac{6}{5}}}{5})$

Step 1.2.10

Multiply.

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Step 1.2.10.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{4}{5}) \frac{x^{- \frac{6}{5}}}{5}$

Step 1.2.10.2

Multiply $\frac{4}{5}$ by $1$ .

$\frac{4}{5} \cdot \frac{x^{- \frac{6}{5}}}{5}$

Step 1.2.11

Multiply $\frac{4}{5}$ by $\frac{x^{- \frac{6}{5}}}{5}$ .

$\frac{4 x^{- \frac{6}{5}}}{5 \cdot 5}$

Step 1.2.12

Multiply.

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Step 1.2.12.1

Multiply $5$ by $5$ .

$\frac{4 x^{- \frac{6}{5}}}{25}$

Step 1.2.12.2

Move $x^{- \frac{6}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{4}{25 x^{\frac{6}{5}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4}{25 x^{\frac{6}{5}}}$ .

$\frac{4}{25 x^{\frac{6}{5}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{4}{25 x^{\frac{6}{5}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{4}{25 x^{\frac{6}{5}}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 = 0$

Step 2.3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 3

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

No Inflection Points