Calculus Examples

$\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$

Step 1

Write $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$ as a function.

$f (x) = \frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}] + \frac{d}{d x} [1]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}]$ .

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Step 2.2.1

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [{(5 - 9 x)}^{\frac{2}{3}}]$ .

$\frac{1}{7} \frac{d}{d x} [{(5 - 9 x)}^{\frac{2}{3}}] + \frac{d}{d x} [1]$

Step 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 $f (x) = x^{\frac{2}{3}}$ and $g (x) = 5 - 9 x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $5 - 9 x$ .

$\frac{1}{7} (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 2.2.2.2

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

$\frac{1}{7} (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 2.2.2.3

Replace all occurrences of $u$ with $5 - 9 x$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 2.2.3

By the Sum Rule, the derivative of $5 - 9 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 9 x]$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (\frac{d}{d x} [5] + \frac{d}{d x} [- 9 x])) + \frac{d}{d x} [1]$

Step 2.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 + \frac{d}{d x} [- 9 x])) + \frac{d}{d x} [1]$

Step 2.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 - 9 \frac{d}{d x} [x])) + \frac{d}{d x} [1]$

Step 2.2.6

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.7

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.8

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.9

Combine the numerators over the common denominator.

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2 - 1 \cdot 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.10

Simplify the numerator.

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Step 2.2.10.1

Multiply $- 1$ by $3$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2 - 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.10.2

Subtract $3$ from $2$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{- 1}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.11

Move the negative in front of the fraction.

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 2.2.12

Multiply $- 9$ by $1$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} (0 - 9)) + \frac{d}{d x} [1]$

Step 2.2.13

Subtract $9$ from $0$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} \cdot - 9) + \frac{d}{d x} [1]$

Step 2.2.14

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

$\frac{1}{7} (\frac{2 {(5 - 9 x)}^{- \frac{1}{3}}}{3} \cdot - 9) + \frac{d}{d x} [1]$

Step 2.2.15

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

$\frac{1}{7} \cdot \frac{2 {(5 - 9 x)}^{- \frac{1}{3}} \cdot - 9}{3} + \frac{d}{d x} [1]$

Step 2.2.16

Multiply $- 9$ by $2$ .

$\frac{1}{7} \cdot \frac{- 18 {(5 - 9 x)}^{- \frac{1}{3}}}{3} + \frac{d}{d x} [1]$

Step 2.2.17

Move ${(5 - 9 x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} \cdot \frac{- 18}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 2.2.18

Factor $3$ out of $- 18$ .

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 2.2.19

Cancel the common factors.

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Step 2.2.19.1

Factor $3$ out of $3 {(5 - 9 x)}^{\frac{1}{3}}$ .

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 ({(5 - 9 x)}^{\frac{1}{3}})} + \frac{d}{d x} [1]$

Step 2.2.19.2

Cancel the common factor.

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 2.2.19.3

Rewrite the expression.

$\frac{1}{7} \cdot \frac{- 6}{{(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 2.2.20

Move the negative in front of the fraction.

$\frac{1}{7} (- \frac{6}{{(5 - 9 x)}^{\frac{1}{3}}}) + \frac{d}{d x} [1]$

Step 2.2.21

Multiply $\frac{1}{7}$ by $\frac{6}{{(5 - 9 x)}^{\frac{1}{3}}}$ .

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} + 0$

Step 2.3.2

Add $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ and $0$ .

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $- \frac{6}{7}$ is constant with respect to $x$ , the derivative of $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ with respect to $x$ is $- \frac{6}{7} \frac{d}{d x} [\frac{1}{{(5 - 9 x)}^{\frac{1}{3}}}]$ .

$f'' (x) = - \frac{6}{7} \cdot \frac{d}{d x} \frac{1}{{(5 - 9 x)}^{\frac{1}{3}}}$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(5 - 9 x)}^{\frac{1}{3}}}$ as ${({(5 - 9 x)}^{\frac{1}{3}})}^{- 1}$ .

$f'' (x) = - \frac{6}{7} \cdot \frac{d}{d x} {({(5 - 9 x)}^{\frac{1}{3}})}^{- 1}$

Step 3.1.2.2

Multiply the exponents in ${({(5 - 9 x)}^{\frac{1}{3}})}^{- 1}$ .

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Step 3.1.2.2.1

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

$f'' (x) = - \frac{6}{7} \cdot \frac{d}{d x} {(5 - 9 x)}^{\frac{1}{3} \cdot - 1}$

Step 3.1.2.2.2

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

$f'' (x) = - \frac{6}{7} \cdot \frac{d}{d x} {(5 - 9 x)}^{\frac{- 1}{3}}$

Step 3.1.2.2.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{7} \cdot \frac{d}{d x} {(5 - 9 x)}^{- \frac{1}{3}}$

Step 3.2

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 $g (x) = 5 - 9 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $5 - 9 x$ .

$f'' (x) = - \frac{6}{7} \cdot (\frac{d}{d u} (u^{- \frac{1}{3}}) \frac{d}{d x} (5 - 9 x))$

Step 3.2.2

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

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} (5 - 9 x))$

Step 3.2.3

Replace all occurrences of $u$ with $5 - 9 x$ .

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{- \frac{1}{3} - 1} \frac{d}{d x} (5 - 9 x))$

Step 3.3

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

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.4

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

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.6.2

Subtract $3$ from $- 1$ .

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{\frac{- 4}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.7

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3} \cdot {(5 - 9 x)}^{- \frac{4}{3}} \frac{d}{d x} (5 - 9 x))$

Step 3.8

Combine ${(5 - 9 x)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$f'' (x) = - \frac{6}{7} \cdot (- \frac{{(5 - 9 x)}^{- \frac{4}{3}}}{3} \cdot \frac{d}{d x} (5 - 9 x))$

Step 3.9

Simplify the expression.

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Step 3.9.1

Move ${(5 - 9 x)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{6}{7} \cdot (- \frac{1}{3 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x))$

Step 3.9.2

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{6}{7}) \cdot (\frac{1}{3 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x))$

Step 3.9.3

Multiply $\frac{6}{7}$ by $1$ .

$f'' (x) = \frac{6}{7} \cdot (\frac{1}{3 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x))$

Step 3.10

Multiply $\frac{1}{3 {(5 - 9 x)}^{\frac{4}{3}}}$ by $\frac{6}{7}$ .

$f'' (x) = \frac{6}{3 {(5 - 9 x)}^{\frac{4}{3}} \cdot 7} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.11

Multiply $7$ by $3$ .

$f'' (x) = \frac{6}{21 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.12

Factor $3$ out of $6$ .

$f'' (x) = \frac{3 (2)}{21 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $3$ out of $21 {(5 - 9 x)}^{\frac{4}{3}}$ .

$f'' (x) = \frac{3 (2)}{3 (7 {(5 - 9 x)}^{\frac{4}{3}})} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.13.2

Cancel the common factor.

$f'' (x) = \frac{3 \cdot 2}{3 (7 {(5 - 9 x)}^{\frac{4}{3}})} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.13.3

Rewrite the expression.

$f'' (x) = \frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (5 - 9 x)$

Step 3.14

By the Sum Rule, the derivative of $5 - 9 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 9 x]$ .

$f'' (x) = \frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot (\frac{d}{d x} (5) + \frac{d}{d x} (- 9 x))$

Step 3.15

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = \frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot (0 + \frac{d}{d x} (- 9 x))$

Step 3.16

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

$f'' (x) = \frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (- 9 x)$

Step 3.17

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot (- 9 \frac{d}{d x} x)$

Step 3.18

Combine fractions.

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Step 3.18.1

Combine $- 9$ and $\frac{2}{7 {(5 - 9 x)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{- 9 \cdot 2}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (x)$

Step 3.18.2

Simplify the expression.

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Step 3.18.2.1

Multiply $- 9$ by $2$ .

$f'' (x) = \frac{- 18}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} (x)$

Step 3.18.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{18}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot \frac{d}{d x} x$

Step 3.19

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

$f'' (x) = - \frac{18}{7 {(5 - 9 x)}^{\frac{4}{3}}} \cdot 1$

Step 3.20

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{18}{7 {(5 - 9 x)}^{\frac{4}{3}}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}] + \frac{d}{d x} [1]$

Step 5.1.2

Evaluate $\frac{d}{d x} [\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}]$ .

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Step 5.1.2.1

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [{(5 - 9 x)}^{\frac{2}{3}}]$ .

$\frac{1}{7} \frac{d}{d x} [{(5 - 9 x)}^{\frac{2}{3}}] + \frac{d}{d x} [1]$

Step 5.1.2.2

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 $g (x) = 5 - 9 x$ .

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Step 5.1.2.2.1

To apply the Chain Rule, set $u$ as $5 - 9 x$ .

$\frac{1}{7} (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 5.1.2.2.2

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

$\frac{1}{7} (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 5.1.2.2.3

Replace all occurrences of $u$ with $5 - 9 x$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} \frac{d}{d x} [5 - 9 x]) + \frac{d}{d x} [1]$

Step 5.1.2.3

By the Sum Rule, the derivative of $5 - 9 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 9 x]$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (\frac{d}{d x} [5] + \frac{d}{d x} [- 9 x])) + \frac{d}{d x} [1]$

Step 5.1.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 + \frac{d}{d x} [- 9 x])) + \frac{d}{d x} [1]$

Step 5.1.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 - 9 \frac{d}{d x} [x])) + \frac{d}{d x} [1]$

Step 5.1.2.6

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.7

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.8

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

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.9

Combine the numerators over the common denominator.

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2 - 1 \cdot 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.10

Simplify the numerator.

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Step 5.1.2.10.1

Multiply $- 1$ by $3$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{2 - 3}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.10.2

Subtract $3$ from $2$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{\frac{- 1}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.11

Move the negative in front of the fraction.

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} (0 - 9 \cdot 1)) + \frac{d}{d x} [1]$

Step 5.1.2.12

Multiply $- 9$ by $1$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} (0 - 9)) + \frac{d}{d x} [1]$

Step 5.1.2.13

Subtract $9$ from $0$ .

$\frac{1}{7} (\frac{2}{3} {(5 - 9 x)}^{- \frac{1}{3}} \cdot - 9) + \frac{d}{d x} [1]$

Step 5.1.2.14

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

$\frac{1}{7} (\frac{2 {(5 - 9 x)}^{- \frac{1}{3}}}{3} \cdot - 9) + \frac{d}{d x} [1]$

Step 5.1.2.15

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

$\frac{1}{7} \cdot \frac{2 {(5 - 9 x)}^{- \frac{1}{3}} \cdot - 9}{3} + \frac{d}{d x} [1]$

Step 5.1.2.16

Multiply $- 9$ by $2$ .

$\frac{1}{7} \cdot \frac{- 18 {(5 - 9 x)}^{- \frac{1}{3}}}{3} + \frac{d}{d x} [1]$

Step 5.1.2.17

Move ${(5 - 9 x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} \cdot \frac{- 18}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 5.1.2.18

Factor $3$ out of $- 18$ .

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 5.1.2.19

Cancel the common factors.

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Step 5.1.2.19.1

Factor $3$ out of $3 {(5 - 9 x)}^{\frac{1}{3}}$ .

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 ({(5 - 9 x)}^{\frac{1}{3}})} + \frac{d}{d x} [1]$

Step 5.1.2.19.2

Cancel the common factor.

$\frac{1}{7} \cdot \frac{3 \cdot - 6}{3 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 5.1.2.19.3

Rewrite the expression.

$\frac{1}{7} \cdot \frac{- 6}{{(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 5.1.2.20

Move the negative in front of the fraction.

$\frac{1}{7} (- \frac{6}{{(5 - 9 x)}^{\frac{1}{3}}}) + \frac{d}{d x} [1]$

Step 5.1.2.21

Multiply $\frac{1}{7}$ by $\frac{6}{{(5 - 9 x)}^{\frac{1}{3}}}$ .

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} + \frac{d}{d x} [1]$

Step 5.1.3

Differentiate using the Constant Rule.

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Step 5.1.3.1

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

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} + 0$

Step 5.1.3.2

Add $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ and $0$ .

$f' (x) = - \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ .

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}} = 0$

Step 6.2

Set the numerator equal to zero.

$6 = 0$

Step 6.3

Since $6 \neq 0$ , there are no solutions.

No solution

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{6}{7 \sqrt[3]{{(5 - 9 x)}^{1}}}$

Step 7.1.2

Anything raised to $1$ is the base itself.

$- \frac{6}{7 \sqrt[3]{5 - 9 x}}$

Step 7.2

Set the denominator in $\frac{6}{7 \sqrt[3]{5 - 9 x}}$ equal to $0$ to find where the expression is undefined.

$7 \sqrt[3]{5 - 9 x} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(7 \sqrt[3]{5 - 9 x})}^{3} = 0^{3}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 - 9 x}$ as ${(5 - 9 x)}^{\frac{1}{3}}$ .

${(7 {(5 - 9 x)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Simplify ${(7 {(5 - 9 x)}^{\frac{1}{3}})}^{3}$ .

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Step 7.3.2.2.1.1

Apply the product rule to $7 {(5 - 9 x)}^{\frac{1}{3}}$ .

$7^{3} {({(5 - 9 x)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.2

Raise $7$ to the power of $3$ .

$343 {({(5 - 9 x)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.3

Multiply the exponents in ${({(5 - 9 x)}^{\frac{1}{3}})}^{3}$ .

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Step 7.3.2.2.1.3.1

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

$343 {(5 - 9 x)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1.3.2.1

Cancel the common factor.

$343 {(5 - 9 x)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

$343 {(5 - 9 x)}^{1} = 0^{3}$

Step 7.3.2.2.1.4

Simplify.

$343 (5 - 9 x) = 0^{3}$

Step 7.3.2.2.1.5

Apply the distributive property.

$343 \cdot 5 + 343 (- 9 x) = 0^{3}$

Step 7.3.2.2.1.6

Multiply.

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Step 7.3.2.2.1.6.1

Multiply $343$ by $5$ .

$1715 + 343 (- 9 x) = 0^{3}$

Step 7.3.2.2.1.6.2

Multiply $- 9$ by $343$ .

$1715 - 3087 x = 0^{3}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$1715 - 3087 x = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Subtract $1715$ from both sides of the equation.

$- 3087 x = - 1715$

Step 7.3.3.2

Divide each term in $- 3087 x = - 1715$ by $- 3087$ and simplify.

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Step 7.3.3.2.1

Divide each term in $- 3087 x = - 1715$ by $- 3087$ .

$\frac{- 3087 x}{- 3087} = \frac{- 1715}{- 3087}$

Step 7.3.3.2.2

Simplify the left side.

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Step 7.3.3.2.2.1

Cancel the common factor of $- 3087$ .

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Step 7.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 3087 x}{- 3087} = \frac{- 1715}{- 3087}$

Step 7.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1715}{- 3087}$

Step 7.3.3.2.3

Simplify the right side.

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Step 7.3.3.2.3.1

Cancel the common factor of $- 1715$ and $- 3087$ .

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Step 7.3.3.2.3.1.1

Factor $- 343$ out of $- 1715$ .

$x = \frac{- 343 (5)}{- 3087}$

Step 7.3.3.2.3.1.2

Cancel the common factors.

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Step 7.3.3.2.3.1.2.1

Factor $- 343$ out of $- 3087$ .

$x = \frac{- 343 \cdot 5}{- 343 \cdot 9}$

Step 7.3.3.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 343 \cdot 5}{- 343 \cdot 9}$

Step 7.3.3.2.3.1.2.3

Rewrite the expression.

$x = \frac{5}{9}$

Step 8

Critical points to evaluate.

$x = \frac{5}{9}$

Step 9

Evaluate the second derivative at $x = \frac{5}{9}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{18}{7 {(5 - 9 (\frac{5}{9}))}^{\frac{4}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Cancel the common factor of $9$ .

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Step 10.1.1.1

Factor $9$ out of $- 9$ .

$- \frac{18}{7 {(5 + 9 (- 1) \frac{5}{9})}^{\frac{4}{3}}}$

Step 10.1.1.2

Cancel the common factor.

$- \frac{18}{7 {(5 + 9 \cdot - 1 \frac{5}{9})}^{\frac{4}{3}}}$

Step 10.1.1.3

Rewrite the expression.

$- \frac{18}{7 {(5 - 1 \cdot 5)}^{\frac{4}{3}}}$

Step 10.1.2

Multiply $- 1$ by $5$ .

$- \frac{18}{7 {(5 - 5)}^{\frac{4}{3}}}$

Step 10.2

Reduce the expression by cancelling the common factors.

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Step 10.2.1

Subtract $5$ from $5$ .

$- \frac{18}{7 \cdot 0^{\frac{4}{3}}}$

Step 10.2.2

Simplify the expression.

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Step 10.2.2.1

Rewrite $0$ as $0^{3}$ .

$- \frac{18}{7 \cdot {(0^{3})}^{\frac{4}{3}}}$

Step 10.2.2.2

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

$- \frac{18}{7 \cdot 0^{3 (\frac{4}{3})}}$

Step 10.2.3

Cancel the common factor of $3$ .

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Step 10.2.3.1

Cancel the common factor.

$- \frac{18}{7 \cdot 0^{3 (\frac{4}{3})}}$

Step 10.2.3.2

Rewrite the expression.

$- \frac{18}{7 \cdot 0^{4}}$

Step 10.2.4

Simplify the expression.

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Step 10.2.4.1

Raising $0$ to any positive power yields $0$ .

$- \frac{18}{7 \cdot 0}$

Step 10.2.4.2

Multiply $7$ by $0$ .

$- \frac{18}{0}$

Step 10.2.4.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{18}{0}$

Step 10.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{18}{0}$

Step 10.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 11

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 11.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \frac{5}{9}) \cup (\frac{5}{9}, \infty)$

Step 11.2

Substitute any number, such as $0$ , from the interval $(- \infty, \frac{5}{9})$ in the first derivative $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 11.2.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = - \frac{6}{7 {(5 - 9 \cdot 0)}^{\frac{1}{3}}}$

Step 11.2.2

Simplify the result.

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Step 11.2.2.1

Simplify the denominator.

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Step 11.2.2.1.1

Multiply $- 9$ by $0$ .

$f' (0) = - \frac{6}{7 {(5 + 0)}^{\frac{1}{3}}}$

Step 11.2.2.1.2

Add $5$ and $0$ .

$f' (0) = - \frac{6}{7 \cdot 5^{\frac{1}{3}}}$

Step 11.2.2.2

The final answer is $- \frac{6}{7 \cdot 5^{\frac{1}{3}}}$ .

$- \frac{6}{7 \cdot 5^{\frac{1}{3}}}$

Step 11.3

Substitute any number, such as $3$ , from the interval $(\frac{5}{9}, \infty)$ in the first derivative $- \frac{6}{7 {(5 - 9 x)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 11.3.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = - \frac{6}{7 {(5 - 9 \cdot 3)}^{\frac{1}{3}}}$

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

Simplify the denominator.

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Step 11.3.2.1.1

Multiply $- 9$ by $3$ .

$f' (3) = - \frac{6}{7 {(5 - 27)}^{\frac{1}{3}}}$

Step 11.3.2.1.2

Subtract $27$ from $5$ .

$f' (3) = - \frac{6}{7 {(- 22)}^{\frac{1}{3}}}$

Step 11.3.2.2

The final answer is $- \frac{6}{7 {(- 22)}^{\frac{1}{3}}}$ .

$- \frac{6}{7 {(- 22)}^{\frac{1}{3}}}$

Step 11.4

Since the first derivative changed signs from negative to positive around $x = \frac{5}{9}$ , then $x = \frac{5}{9}$ is a local minimum.

$x = \frac{5}{9}$ is a local minimum

Step 12