Calculus Examples

$f (x) = \sqrt[3]{x - 1}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1}$ as ${(x - 1)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(x - 1)}^{\frac{1}{3}}]$

Step 1.1.2

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

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Replace all occurrences of $u$ with $x - 1$ .

$\frac{1}{3} {(x - 1)}^{\frac{1}{3} - 1} \frac{d}{d x} [x - 1]$

Step 1.1.3

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

$\frac{1}{3} {(x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 1]$

Step 1.1.4

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

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

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{3} {(x - 1)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 1]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(x - 1)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x - 1]$

Step 1.1.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(x - 1)}^{\frac{- 2}{3}} \frac{d}{d x} [x - 1]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(x - 1)}^{- \frac{2}{3}} \frac{d}{d x} [x - 1]$

Step 1.1.7.2

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

$\frac{{(x - 1)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x - 1]$

Step 1.1.7.3

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

$\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \frac{d}{d x} [x - 1]$

Step 1.1.8

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

$\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 1.1.9

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

$\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} (1 + \frac{d}{d x} [- 1])$

Step 1.1.10

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

$\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} (1 + 0)$

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Add $1$ and $0$ .

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

Step 1.1.11.2

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

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate using the Constant Multiple Rule.

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Step 1.2.1.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{1}{{(x - 1)}^{\frac{2}{3}}}]$ .

$\frac{1}{3} \frac{d}{d x} [\frac{1}{{(x - 1)}^{\frac{2}{3}}}]$

Step 1.2.1.2

Apply basic rules of exponents.

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Step 1.2.1.2.1

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

$\frac{1}{3} \frac{d}{d x} [{({(x - 1)}^{\frac{2}{3}})}^{- 1}]$

Step 1.2.1.2.2

Multiply the exponents in ${({(x - 1)}^{\frac{2}{3}})}^{- 1}$ .

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Step 1.2.1.2.2.1

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

$\frac{1}{3} \frac{d}{d x} [{(x - 1)}^{\frac{2}{3} \cdot - 1}]$

Step 1.2.1.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 1.2.1.2.2.2.1

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

$\frac{1}{3} \frac{d}{d x} [{(x - 1)}^{\frac{2 \cdot - 1}{3}}]$

Step 1.2.1.2.2.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} \frac{d}{d x} [{(x - 1)}^{\frac{- 2}{3}}]$

Step 1.2.1.2.2.3

Move the negative in front of the fraction.

$\frac{1}{3} \frac{d}{d x} [{(x - 1)}^{- \frac{2}{3}}]$

Step 1.2.2

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

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$\frac{1}{3} (\frac{d}{d u} [u^{- \frac{2}{3}}] \frac{d}{d x} [x - 1])$

Step 1.2.2.2

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

$\frac{1}{3} (- \frac{2}{3} u^{- \frac{2}{3} - 1} \frac{d}{d x} [x - 1])$

Step 1.2.2.3

Replace all occurrences of $u$ with $x - 1$ .

$\frac{1}{3} (- \frac{2}{3} {(x - 1)}^{- \frac{2}{3} - 1} \frac{d}{d x} [x - 1])$

Step 1.2.3

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

$\frac{1}{3} (- \frac{2}{3} {(x - 1)}^{- \frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 1])$

Step 1.2.4

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

$\frac{1}{3} (- \frac{2}{3} {(x - 1)}^{- \frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 1])$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{1}{3} (- \frac{2}{3} {(x - 1)}^{\frac{- 2 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 1])$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} (- \frac{2}{3} {(x - 1)}^{\frac{- 2 - 3}{3}} \frac{d}{d x} [x - 1])$

Step 1.2.6.2

Subtract $3$ from $- 2$ .

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

Step 1.2.7

Combine fractions.

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Step 1.2.7.1

Move the negative in front of the fraction.

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

Step 1.2.7.2

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

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

Step 1.2.7.3

Simplify the expression.

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Step 1.2.7.3.1

Move $2$ to the left of ${(x - 1)}^{- \frac{5}{3}}$ .

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

Step 1.2.7.3.2

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

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

Step 1.2.7.4

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

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

Step 1.2.7.5

Multiply $3$ by $3$ .

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} \frac{d}{d x} [x - 1]$

Step 1.2.8

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

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 1.2.9

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

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} (1 + \frac{d}{d x} [- 1])$

Step 1.2.10

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

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} (1 + 0)$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Add $1$ and $0$ .

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} \cdot 1$

Step 1.2.11.2

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{2}{9 {(x - 1)}^{\frac{5}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}}$ .

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{2}{9 {(x - 1)}^{\frac{5}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

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

No solution

Step 3

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

No Inflection Points