Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

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

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Step 1.2.2.1

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

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

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 $n = \frac{1}{3}$ .

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

Step 1.2.2.3

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

Combine the numerators over the common denominator.

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

Step 1.2.10

Simplify the numerator.

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Step 1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 1.2.10.2

Subtract $3$ from $1$ .

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

Step 1.2.11

Move the negative in front of the fraction.

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

Step 1.2.12

Multiply $- 1$ by $1$ .

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

Step 1.2.13

Subtract $1$ from $0$ .

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

Step 1.2.14

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

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

Step 1.2.15

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

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

Step 1.2.16

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

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

Step 1.2.17

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

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

Step 1.2.18

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

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

Step 1.2.19

Move the negative in front of the fraction.

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

Step 1.2.20

Multiply $- 1$ by $3$ .

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

Step 1.2.21

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

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

Step 1.2.22

Factor $3$ out of $- 3$ .

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

Step 1.2.23

Cancel the common factors.

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Step 1.2.23.1

Factor $3$ out of $3 {(1 - x)}^{\frac{2}{3}}$ .

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

Step 1.2.23.2

Cancel the common factor.

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

Step 1.2.23.3

Rewrite the expression.

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

Step 1.2.24

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 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 $g (x) = \frac{1}{{(1 - x)}^{\frac{2}{3}}}$ .

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

Step 2.2.2

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

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

Step 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 $g (x) = {(1 - x)}^{\frac{2}{3}}$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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 $g (x) = 1 - x$ .

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

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

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Step 2.2.10.1

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

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

Step 2.2.10.2

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

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Step 2.2.10.2.1

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

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

Step 2.2.10.2.2

Multiply $2$ by $- 2$ .

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

Step 2.2.10.3

Move the negative in front of the fraction.

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

Step 2.2.11

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

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

Step 2.2.12

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

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

Step 2.2.13

Combine the numerators over the common denominator.

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

Step 2.2.14

Simplify the numerator.

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Step 2.2.14.1

Multiply $- 1$ by $3$ .

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

Step 2.2.14.2

Subtract $3$ from $2$ .

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

Step 2.2.15

Move the negative in front of the fraction.

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

Step 2.2.16

Multiply $- 1$ by $1$ .

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

Step 2.2.17

Subtract $1$ from $0$ .

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

Step 2.2.18

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

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

Step 2.2.19

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

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

Step 2.2.20

Multiply $- 1$ by $2$ .

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

Step 2.2.21

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

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

Step 2.2.22

Move the negative in front of the fraction.

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

Step 2.2.23

Multiply $- 1$ by $- 1$ .

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

Step 2.2.24

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

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

Step 2.2.25

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

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

Step 2.2.26

Move $2$ to the left of ${(1 - x)}^{- \frac{4}{3}}$ .

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

Step 2.2.27

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

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

Step 2.2.28

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

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Step 2.2.28.1

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

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

Step 2.2.28.2

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

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

Step 2.2.28.3

Combine the numerators over the common denominator.

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

Step 2.2.28.4

Add $4$ and $1$ .

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

Step 2.2.29

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

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

Step 2.2.30

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

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

Step 2.3

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

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

Step 3

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

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