Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

Combine the numerators over the common denominator.

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

Step 1.1.2.9

Simplify the numerator.

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Step 1.1.2.9.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.9.2

Subtract $3$ from $1$ .

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

Step 1.1.2.10

Move the negative in front of the fraction.

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

Step 1.1.2.11

Add $1$ and $0$ .

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

Step 1.1.2.12

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

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

Step 1.1.2.13

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

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

Step 1.1.2.14

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

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

Step 1.1.2.15

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

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

Step 1.1.2.16

Cancel the common factor.

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

Step 1.1.2.17

Rewrite the expression.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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Step 1.2.2.2.1

To apply the Chain Rule, set $u_{1}$ as ${(x - 1)}^{\frac{2}{3}}$ .

$0 + \frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [{(x - 1)}^{\frac{2}{3}}]$

Step 1.2.2.2.2

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

$0 - {u_{1}}^{- 2} \frac{d}{d x} [{(x - 1)}^{\frac{2}{3}}]$

Step 1.2.2.2.3

Replace all occurrences of $u_{1}$ with ${(x - 1)}^{\frac{2}{3}}$ .

$0 - {({(x - 1)}^{\frac{2}{3}})}^{- 2} \frac{d}{d x} [{(x - 1)}^{\frac{2}{3}}]$

Step 1.2.2.3

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

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Step 1.2.2.3.1

To apply the Chain Rule, set $u_{2}$ as $x - 1$ .

$0 - {({(x - 1)}^{\frac{2}{3}})}^{- 2} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{2}{3}}] \frac{d}{d x} [x - 1])$

Step 1.2.2.3.2

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

$0 - {({(x - 1)}^{\frac{2}{3}})}^{- 2} (\frac{2}{3} {u_{2}}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1])$

Step 1.2.2.3.3

Replace all occurrences of $u_{2}$ with $x - 1$ .

$0 - {({(x - 1)}^{\frac{2}{3}})}^{- 2} (\frac{2}{3} {(x - 1)}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1])$

Step 1.2.2.4

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

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

Step 1.2.2.5

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

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

Step 1.2.2.6

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

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

Step 1.2.2.7

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

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Step 1.2.2.7.1

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

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

Step 1.2.2.7.2

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

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Step 1.2.2.7.2.1

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

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

Step 1.2.2.7.2.2

Multiply $2$ by $- 2$ .

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

Step 1.2.2.7.3

Move the negative in front of the fraction.

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

Step 1.2.2.8

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

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

Step 1.2.2.9

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

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

Step 1.2.2.10

Combine the numerators over the common denominator.

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

Step 1.2.2.11

Simplify the numerator.

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Step 1.2.2.11.1

Multiply $- 1$ by $3$ .

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

Step 1.2.2.11.2

Subtract $3$ from $2$ .

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

Step 1.2.2.12

Move the negative in front of the fraction.

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

Step 1.2.2.13

Add $1$ and $0$ .

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

Step 1.2.2.14

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

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

Step 1.2.2.15

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

$0 - {(x - 1)}^{- \frac{4}{3}} \frac{2 {(x - 1)}^{- \frac{1}{3}}}{3}$

Step 1.2.2.16

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

$0 - {(x - 1)}^{- \frac{4}{3}} \frac{2}{3 {(x - 1)}^{\frac{1}{3}}}$

Step 1.2.2.17

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

$0 - \frac{2 {(x - 1)}^{- \frac{4}{3}}}{3 {(x - 1)}^{\frac{1}{3}}}$

Step 1.2.2.18

Move ${(x - 1)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{2}{3 {(x - 1)}^{\frac{1}{3}} {(x - 1)}^{\frac{4}{3}}}$

Step 1.2.2.19

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

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Step 1.2.2.19.1

Move ${(x - 1)}^{\frac{4}{3}}$ .

$0 - \frac{2}{3 ({(x - 1)}^{\frac{4}{3}} {(x - 1)}^{\frac{1}{3}})}$

Step 1.2.2.19.2

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

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

Step 1.2.2.19.3

Combine the numerators over the common denominator.

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

Step 1.2.2.19.4

Add $4$ and $1$ .

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

Step 1.2.3

Subtract $\frac{2}{3 {(x - 1)}^{\frac{5}{3}}}$ from $0$ .

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

Step 1.3

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

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

Step 2

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

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Step 2.1

Set the second derivative equal to $0$ .

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

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