Calculus Examples

$y = x - 3 x^{\frac{1}{3}}$

Step 1

Write $y = x - 3 x^{\frac{1}{3}}$ as a function.

$f (x) = x - 3 x^{\frac{1}{3}}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

By the Sum Rule, the derivative of $x - 3 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3 x^{\frac{1}{3}}]$ .

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (- 3 x^{\frac{1}{3}})$

Step 2.1.1.2

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

$f' (x) = 1 + \frac{d}{d x} (- 3 x^{\frac{1}{3}})$

Step 2.1.2

Evaluate $\frac{d}{d x} [- 3 x^{\frac{1}{3}}]$ .

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Step 2.1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{\frac{1}{3}}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$f' (x) = 1 - 3 \frac{d}{d x} x^{\frac{1}{3}}$

Step 2.1.2.2

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

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

Step 2.1.2.3

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

$f' (x) = 1 - 3 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 2.1.2.4

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

$f' (x) = 1 - 3 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.1.2.5

Combine the numerators over the common denominator.

$f' (x) = 1 - 3 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 2.1.2.6

Simplify the numerator.

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Step 2.1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.6.2

Subtract $3$ from $1$ .

$f' (x) = 1 - 3 (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 2.1.2.7

Move the negative in front of the fraction.

$f' (x) = 1 - 3 (\frac{1}{3} x^{- \frac{2}{3}})$

Step 2.1.2.8

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

$f' (x) = 1 - 3 \frac{x^{- \frac{2}{3}}}{3}$

Step 2.1.2.9

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

$f' (x) = 1 + \frac{- 3 x^{- \frac{2}{3}}}{3}$

Step 2.1.2.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = 1 + \frac{- 3}{3 x^{\frac{2}{3}}}$

Step 2.1.2.11

Factor $3$ out of $- 3$ .

$f' (x) = 1 + \frac{3 \cdot - 1}{3 x^{\frac{2}{3}}}$

Step 2.1.2.12

Cancel the common factors.

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Step 2.1.2.12.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

$f' (x) = 1 + \frac{3 \cdot - 1}{3 (x^{\frac{2}{3}})}$

Step 2.1.2.12.2

Cancel the common factor.

$f' (x) = 1 + \frac{3 \cdot - 1}{3 x^{\frac{2}{3}}}$

Step 2.1.2.12.3

Rewrite the expression.

$f' (x) = 1 + \frac{- 1}{x^{\frac{2}{3}}}$

Step 2.1.2.13

Move the negative in front of the fraction.

$f' (x) = 1 - \frac{1}{x^{\frac{2}{3}}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

By the Sum Rule, the derivative of $1 - \frac{1}{x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{1}{x^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- \frac{1}{x^{\frac{2}{3}}})$

Step 2.2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \frac{1}{x^{\frac{2}{3}}})$

Step 2.2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x^{\frac{2}{3}}}]$ .

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Step 2.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{2}{3}}}$ .

$f'' (x) = 0 - \frac{d}{d x} \cdot \frac{1}{x^{\frac{2}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

$f'' (x) = 0 - \frac{d}{d x} {(x^{\frac{2}{3}})}^{- 1} + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{2}{3}}$ .

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Step 2.2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

$f'' (x) = 0 - (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{2}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.3.2

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

$f'' (x) = 0 - (- u^{- 2} \frac{d}{d x} x^{\frac{2}{3}}) + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$f'' (x) = 0 - (- {(x^{\frac{2}{3}})}^{- 2} \frac{d}{d x} x^{\frac{2}{3}}) + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.4

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

$f'' (x) = 0 - (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.2.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = 0 - (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.6

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 2}$ .

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Step 2.2.2.6.1

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

$f'' (x) = 0 - (- x^{\frac{2}{3} \cdot - 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.6.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 2.2.2.6.2.1

Combine $\frac{2}{3}$ and $- 2$ .

$f'' (x) = 0 - (- x^{\frac{2 \cdot - 2}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.6.2.2

Multiply $2$ by $- 2$ .

$f'' (x) = 0 - (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.6.3

Move the negative in front of the fraction.

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.7

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

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.8

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

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.9

Combine the numerators over the common denominator.

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.10

Simplify the numerator.

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Step 2.2.2.10.1

Multiply $- 1$ by $3$ .

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.10.2

Subtract $3$ from $2$ .

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.11

Move the negative in front of the fraction.

$f'' (x) = 0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.12

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

$f'' (x) = 0 - (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3}) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.13

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$f'' (x) = 0 + \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.14

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 2.2.2.14.1

Move $x^{- \frac{4}{3}}$ .

$f'' (x) = 0 + \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.14.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 0 + \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.14.3

Combine the numerators over the common denominator.

$f'' (x) = 0 + \frac{2 x^{\frac{- 4 - 1}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.14.4

Subtract $1$ from $- 4$ .

$f'' (x) = 0 + \frac{2 x^{\frac{- 5}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.14.5

Move the negative in front of the fraction.

$f'' (x) = 0 + \frac{2 x^{- \frac{5}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.15

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 + \frac{2}{3 x^{\frac{5}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.16

Multiply $- 1$ by $- 1$ .

$f'' (x) = 0 + 1 (\frac{2}{3 x^{\frac{5}{3}}}) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.17

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

$f'' (x) = 0 + \frac{2}{3 x^{\frac{5}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 2.2.2.18

Multiply $\frac{1}{x^{\frac{2}{3}}}$ by $0$ .

$f'' (x) = 0 + \frac{2}{3 x^{\frac{5}{3}}} + 0$

Step 2.2.2.19

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

$f'' (x) = 0 + \frac{2}{3 x^{\frac{5}{3}}}$

Step 2.2.3

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

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

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2}{3 x^{\frac{5}{3}}}$ .

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

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{2}{3 x^{\frac{5}{3}}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

Set the numerator equal to zero.

$2 = 0$

Step 3.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 4

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

No Inflection Points