Calculus Examples

$f (x) = x^{\frac{5}{3}} + 3$

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

By the Sum Rule, the derivative of $x^{\frac{5}{3}} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{5}{3}}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x^{\frac{5}{3}}] + \frac{d}{d x} [3]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{5}{3}}]$ .

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{3}$ .

$\frac{5}{3} x^{\frac{5}{3} - 1} + \frac{d}{d x} [3]$

Step 1.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} [3]$

Step 1.1.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} [3]$

Step 1.1.2.4

Combine the numerators over the common denominator.

$\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}} + \frac{d}{d x} [3]$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $3$ .

$\frac{5}{3} x^{\frac{5 - 3}{3}} + \frac{d}{d x} [3]$

Step 1.1.2.5.2

Subtract $3$ from $5$ .

$\frac{5}{3} x^{\frac{2}{3}} + \frac{d}{d x} [3]$

Step 1.1.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{5}{3} x^{\frac{2}{3}} + 0$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Add $\frac{5}{3} x^{\frac{2}{3}}$ and $0$ .

$\frac{5}{3} x^{\frac{2}{3}}$

Step 1.1.4.2

Combine $\frac{5}{3}$ and $x^{\frac{2}{3}}$ .

$f' (x) = \frac{5 x^{\frac{2}{3}}}{3}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $\frac{5}{3}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{\frac{2}{3}}}{3}$ with respect to $x$ is $\frac{5}{3} \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$\frac{5}{3} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{5}{3} (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{3} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 1.2.4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{5}{3} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{5}{3} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

$\frac{5}{3} (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 1.2.6.2

Subtract $3$ from $2$ .

$\frac{5}{3} (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 1.2.7

Move the negative in front of the fraction.

$\frac{5}{3} (\frac{2}{3} x^{- \frac{1}{3}})$

Step 1.2.8

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$\frac{5}{3} \cdot \frac{2 x^{- \frac{1}{3}}}{3}$

Step 1.2.9

Multiply $\frac{5}{3}$ by $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{5 (2 x^{- \frac{1}{3}})}{3 \cdot 3}$

Step 1.2.10

Multiply.

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Step 1.2.10.1

Multiply $2$ by $5$ .

$\frac{10 x^{- \frac{1}{3}}}{3 \cdot 3}$

Step 1.2.10.2

Multiply $3$ by $3$ .

$\frac{10 x^{- \frac{1}{3}}}{9}$

Step 1.2.10.3

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{10}{9 x^{\frac{1}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{10}{9 x^{\frac{1}{3}}}$ .

$\frac{10}{9 x^{\frac{1}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{10}{9 x^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{10}{9 x^{\frac{1}{3}}} = 0$

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