Calculus Examples

$f (x) = 5 - {(6 + 5 x)}^{\frac{2}{5}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $5 - {(6 + 5 x)}^{\frac{2}{5}}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$

Step 1.1.2

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

$0 + \frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$

Step 1.2

Evaluate $\frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$ .

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(6 + 5 x)}^{\frac{2}{5}}$ with respect to $x$ is $- \frac{d}{d x} [{(6 + 5 x)}^{\frac{2}{5}}]$ .

$0 - \frac{d}{d x} [{(6 + 5 x)}^{\frac{2}{5}}]$

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 $f (x) = x^{\frac{2}{5}}$ and $g (x) = 6 + 5 x$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $6 + 5 x$ .

$0 - (\frac{d}{d u} [u^{\frac{2}{5}}] \frac{d}{d x} [6 + 5 x])$

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 $n = \frac{2}{5}$ .

$0 - (\frac{2}{5} u^{\frac{2}{5} - 1} \frac{d}{d x} [6 + 5 x])$

Step 1.2.2.3

Replace all occurrences of $u$ with $6 + 5 x$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1} \frac{d}{d x} [6 + 5 x])$

Step 1.2.3

By the Sum Rule, the derivative of $6 + 5 x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [5 x]$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1} (\frac{d}{d x} [6] + \frac{d}{d x} [5 x]))$

Step 1.2.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1} (0 + \frac{d}{d x} [5 x]))$

Step 1.2.5

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1} (0 + 5 \frac{d}{d x} [x]))$

Step 1.2.6

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

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1} (0 + 5 \cdot 1))$

Step 1.2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} - 1 \cdot \frac{5}{5}} (0 + 5 \cdot 1))$

Step 1.2.8

Combine $- 1$ and $\frac{5}{5}$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}} (0 + 5 \cdot 1))$

Step 1.2.9

Combine the numerators over the common denominator.

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2 - 1 \cdot 5}{5}} (0 + 5 \cdot 1))$

Step 1.2.10

Simplify the numerator.

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Step 1.2.10.1

Multiply $- 1$ by $5$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{2 - 5}{5}} (0 + 5 \cdot 1))$

Step 1.2.10.2

Subtract $5$ from $2$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{\frac{- 3}{5}} (0 + 5 \cdot 1))$

Step 1.2.11

Move the negative in front of the fraction.

$0 - (\frac{2}{5} {(6 + 5 x)}^{- \frac{3}{5}} (0 + 5 \cdot 1))$

Step 1.2.12

Multiply $5$ by $1$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{- \frac{3}{5}} (0 + 5))$

Step 1.2.13

Add $0$ and $5$ .

$0 - (\frac{2}{5} {(6 + 5 x)}^{- \frac{3}{5}} \cdot 5)$

Step 1.2.14

Combine $\frac{2}{5}$ and ${(6 + 5 x)}^{- \frac{3}{5}}$ .

$0 - (\frac{2 {(6 + 5 x)}^{- \frac{3}{5}}}{5} \cdot 5)$

Step 1.2.15

Combine $\frac{2 {(6 + 5 x)}^{- \frac{3}{5}}}{5}$ and $5$ .

$0 - \frac{2 {(6 + 5 x)}^{- \frac{3}{5}} \cdot 5}{5}$

Step 1.2.16

Multiply $5$ by $2$ .

$0 - \frac{10 {(6 + 5 x)}^{- \frac{3}{5}}}{5}$

Step 1.2.17

Move ${(6 + 5 x)}^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{10}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 1.2.18

Factor $5$ out of $10$ .

$0 - \frac{5 \cdot 2}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 1.2.19

Cancel the common factors.

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Step 1.2.19.1

Factor $5$ out of $5 {(6 + 5 x)}^{\frac{3}{5}}$ .

$0 - \frac{5 \cdot 2}{5 ({(6 + 5 x)}^{\frac{3}{5}})}$

Step 1.2.19.2

Cancel the common factor.

$0 - \frac{5 \cdot 2}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 1.2.19.3

Rewrite the expression.

$0 - \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 1.3

Subtract $\frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ from $0$ .

$- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{{(6 + 5 x)}^{\frac{3}{5}}}]$ .

$f'' (x) = - 2 \frac{d}{d x} \frac{1}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(6 + 5 x)}^{\frac{3}{5}}}$ as ${({(6 + 5 x)}^{\frac{3}{5}})}^{- 1}$ .

$f'' (x) = - 2 \frac{d}{d x} {({(6 + 5 x)}^{\frac{3}{5}})}^{- 1}$

Step 2.1.2.2

Multiply the exponents in ${({(6 + 5 x)}^{\frac{3}{5}})}^{- 1}$ .

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Step 2.1.2.2.1

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

$f'' (x) = - 2 \frac{d}{d x} {(6 + 5 x)}^{\frac{3}{5} \cdot - 1}$

Step 2.1.2.2.2

Multiply $\frac{3}{5} \cdot - 1$ .

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Step 2.1.2.2.2.1

Combine $\frac{3}{5}$ and $- 1$ .

$f'' (x) = - 2 \frac{d}{d x} {(6 + 5 x)}^{\frac{3 \cdot - 1}{5}}$

Step 2.1.2.2.2.2

Multiply $3$ by $- 1$ .

$f'' (x) = - 2 \frac{d}{d x} {(6 + 5 x)}^{\frac{- 3}{5}}$

Step 2.1.2.2.3

Move the negative in front of the fraction.

$f'' (x) = - 2 \frac{d}{d x} {(6 + 5 x)}^{- \frac{3}{5}}$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{3}{5}}$ and $g (x) = 6 + 5 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $6 + 5 x$ .

$f'' (x) = - 2 (\frac{d}{d u} (u^{- \frac{3}{5}}) \frac{d}{d x} (6 + 5 x))$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{3}{5}$ .

$f'' (x) = - 2 (- \frac{3}{5} u^{- \frac{3}{5} - 1} \frac{d}{d x} (6 + 5 x))$

Step 2.2.3

Replace all occurrences of $u$ with $6 + 5 x$ .

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{- \frac{3}{5} - 1} \frac{d}{d x} (6 + 5 x))$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{- \frac{3}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.4

Combine $- 1$ and $\frac{5}{5}$ .

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{- \frac{3}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.5

Combine the numerators over the common denominator.

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{\frac{- 3 - 1 \cdot 5}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $5$ .

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{\frac{- 3 - 5}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.6.2

Subtract $5$ from $- 3$ .

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{\frac{- 8}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$f'' (x) = - 2 (- \frac{3}{5} \cdot {(6 + 5 x)}^{- \frac{8}{5}} \frac{d}{d x} (6 + 5 x))$

Step 2.7.2

Combine ${(6 + 5 x)}^{- \frac{8}{5}}$ and $\frac{3}{5}$ .

$f'' (x) = - 2 (- \frac{{(6 + 5 x)}^{- \frac{8}{5}} \cdot 3}{5} \cdot \frac{d}{d x} (6 + 5 x))$

Step 2.7.3

Simplify the expression.

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Step 2.7.3.1

Move $3$ to the left of ${(6 + 5 x)}^{- \frac{8}{5}}$ .

$f'' (x) = - 2 (- \frac{3 \cdot {(6 + 5 x)}^{- \frac{8}{5}}}{5} \cdot \frac{d}{d x} (6 + 5 x))$

Step 2.7.3.2

Move ${(6 + 5 x)}^{- \frac{8}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 2 (- \frac{3}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (6 + 5 x))$

Step 2.7.3.3

Multiply $- 1$ by $- 2$ .

$f'' (x) = 2 (\frac{3}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (6 + 5 x))$

Step 2.7.4

Combine $\frac{3}{5 {(6 + 5 x)}^{\frac{8}{5}}}$ and $2$ .

$f'' (x) = \frac{3 \cdot 2}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (6 + 5 x)$

Step 2.7.5

Multiply $3$ by $2$ .

$f'' (x) = \frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (6 + 5 x)$

Step 2.8

By the Sum Rule, the derivative of $6 + 5 x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [5 x]$ .

$f'' (x) = \frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot (\frac{d}{d x} (6) + \frac{d}{d x} (5 x))$

Step 2.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f'' (x) = \frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot (0 + \frac{d}{d x} (5 x))$

Step 2.10

Add $0$ and $\frac{d}{d x} [5 x]$ .

$f'' (x) = \frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (5 x)$

Step 2.11

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot (5 \frac{d}{d x} (x))$

Step 2.12

Simplify terms.

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Step 2.12.1

Combine $5$ and $\frac{6}{5 {(6 + 5 x)}^{\frac{8}{5}}}$ .

$f'' (x) = \frac{5 \cdot 6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (x)$

Step 2.12.2

Multiply $5$ by $6$ .

$f'' (x) = \frac{30}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (x)$

Step 2.12.3

Factor $5$ out of $30$ .

$f'' (x) = \frac{5 \cdot 6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (x)$

Step 2.13

Cancel the common factors.

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Step 2.13.1

Factor $5$ out of $5 {(6 + 5 x)}^{\frac{8}{5}}$ .

$f'' (x) = \frac{5 \cdot 6}{5 ({(6 + 5 x)}^{\frac{8}{5}})} \cdot \frac{d}{d x} (x)$

Step 2.13.2

Cancel the common factor.

$f'' (x) = \frac{5 \cdot 6}{5 {(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (x)$

Step 2.13.3

Rewrite the expression.

$f'' (x) = \frac{6}{{(6 + 5 x)}^{\frac{8}{5}}} \cdot \frac{d}{d x} (x)$

Step 2.14

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

$f'' (x) = \frac{6}{{(6 + 5 x)}^{\frac{8}{5}}} \cdot 1$

Step 2.15

Simplify the expression.

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Step 2.15.1

Multiply $\frac{6}{{(6 + 5 x)}^{\frac{8}{5}}}$ by $1$ .

$f'' (x) = \frac{6}{{(6 + 5 x)}^{\frac{8}{5}}}$

Step 2.15.2

Reorder terms.

$f'' (x) = \frac{6}{{(5 x + 6)}^{\frac{8}{5}}}$

Step 3

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

$- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $5 - {(6 + 5 x)}^{\frac{2}{5}}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$ .

$f' (x) = \frac{d}{d x} (5) + \frac{d}{d x} (- {(6 + 5 x)}^{\frac{2}{5}})$

Step 4.1.1.2

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

$f' (x) = 0 + \frac{d}{d x} (- {(6 + 5 x)}^{\frac{2}{5}})$

Step 4.1.2

Evaluate $\frac{d}{d x} [- {(6 + 5 x)}^{\frac{2}{5}}]$ .

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Step 4.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(6 + 5 x)}^{\frac{2}{5}}$ with respect to $x$ is $- \frac{d}{d x} [{(6 + 5 x)}^{\frac{2}{5}}]$ .

$f' (x) = 0 - \frac{d}{d x} {(6 + 5 x)}^{\frac{2}{5}}$

Step 4.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{5}}$ and $g (x) = 6 + 5 x$ .

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Step 4.1.2.2.1

To apply the Chain Rule, set $u$ as $6 + 5 x$ .

$f' (x) = 0 - (\frac{d}{d u} (u^{\frac{2}{5}}) \frac{d}{d x} (6 + 5 x))$

Step 4.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{5}$ .

$f' (x) = 0 - (\frac{2}{5} u^{\frac{2}{5} - 1} \frac{d}{d x} (6 + 5 x))$

Step 4.1.2.2.3

Replace all occurrences of $u$ with $6 + 5 x$ .

$f' (x) = 0 - (\frac{2}{5} \cdot {(6 + 5 x)}^{\frac{2}{5} - 1} \frac{d}{d x} (6 + 5 x))$

Step 4.1.2.3

By the Sum Rule, the derivative of $6 + 5 x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [5 x]$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} - 1} (\frac{d}{d x} (6) + \frac{d}{d x} (5 x))))$

Step 4.1.2.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} - 1} (0 + \frac{d}{d x} (5 x))))$

Step 4.1.2.5

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} - 1} (0 + 5 \frac{d}{d x} (x))))$

Step 4.1.2.6

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

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} - 1} (0 + 5 \cdot 1)))$

Step 4.1.2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} - 1 \cdot \frac{5}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.8

Combine $- 1$ and $\frac{5}{5}$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.9

Combine the numerators over the common denominator.

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2 - 1 \cdot 5}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.10

Simplify the numerator.

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Step 4.1.2.10.1

Multiply $- 1$ by $5$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{2 - 5}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.10.2

Subtract $5$ from $2$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{\frac{- 3}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.11

Move the negative in front of the fraction.

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{- \frac{3}{5}} (0 + 5 \cdot 1)))$

Step 4.1.2.12

Multiply $5$ by $1$ .

$f' (x) = 0 - (\frac{2}{5} \cdot ({(6 + 5 x)}^{- \frac{3}{5}} (0 + 5)))$

Step 4.1.2.13

Add $0$ and $5$ .

$f' (x) = 0 - (\frac{2}{5} \cdot {(6 + 5 x)}^{- \frac{3}{5}} \cdot 5)$

Step 4.1.2.14

Combine $\frac{2}{5}$ and ${(6 + 5 x)}^{- \frac{3}{5}}$ .

$f' (x) = 0 - (\frac{2 {(6 + 5 x)}^{- \frac{3}{5}}}{5} \cdot 5)$

Step 4.1.2.15

Combine $\frac{2 {(6 + 5 x)}^{- \frac{3}{5}}}{5}$ and $5$ .

$f' (x) = 0 - \frac{2 {(6 + 5 x)}^{- \frac{3}{5}} \cdot 5}{5}$

Step 4.1.2.16

Multiply $5$ by $2$ .

$f' (x) = 0 - \frac{10 {(6 + 5 x)}^{- \frac{3}{5}}}{5}$

Step 4.1.2.17

Move ${(6 + 5 x)}^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = 0 - \frac{10}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 4.1.2.18

Factor $5$ out of $10$ .

$f' (x) = 0 - \frac{5 \cdot 2}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 4.1.2.19

Cancel the common factors.

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Step 4.1.2.19.1

Factor $5$ out of $5 {(6 + 5 x)}^{\frac{3}{5}}$ .

$f' (x) = 0 - \frac{5 \cdot 2}{5 ({(6 + 5 x)}^{\frac{3}{5}})}$

Step 4.1.2.19.2

Cancel the common factor.

$f' (x) = 0 - \frac{5 \cdot 2}{5 {(6 + 5 x)}^{\frac{3}{5}}}$

Step 4.1.2.19.3

Rewrite the expression.

$f' (x) = 0 - \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 4.1.3

Subtract $\frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ from $0$ .

$f' (x) = - \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ .

$- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}} = 0$

Step 5.2

Set the numerator equal to zero.

$2 = 0$

Step 5.3

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

No solution

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$- \frac{2}{\sqrt[5]{{(6 + 5 x)}^{3}}}$

Step 6.2

Set the denominator in $\frac{2}{\sqrt[5]{{(6 + 5 x)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[5]{{(6 + 5 x)}^{3}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{{(6 + 5 x)}^{3}}}^{5} = 0^{5}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(6 + 5 x)}^{3}}$ as ${(6 + 5 x)}^{\frac{3}{5}}$ .

${({(6 + 5 x)}^{\frac{3}{5}})}^{5} = 0^{5}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Multiply the exponents in ${({(6 + 5 x)}^{\frac{3}{5}})}^{5}$ .

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Step 6.3.2.2.1.1

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

${(6 + 5 x)}^{\frac{3}{5} \cdot 5} = 0^{5}$

Step 6.3.2.2.1.2

Cancel the common factor of $5$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

${(6 + 5 x)}^{\frac{3}{5} \cdot 5} = 0^{5}$

Step 6.3.2.2.1.2.2

Rewrite the expression.

${(6 + 5 x)}^{3} = 0^{5}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

${(6 + 5 x)}^{3} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Set the $6 + 5 x$ equal to $0$ .

$6 + 5 x = 0$

Step 6.3.3.2

Solve for $x$ .

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Step 6.3.3.2.1

Subtract $6$ from both sides of the equation.

$5 x = - 6$

Step 6.3.3.2.2

Divide each term in $5 x = - 6$ by $5$ and simplify.

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Step 6.3.3.2.2.1

Divide each term in $5 x = - 6$ by $5$ .

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 6.3.3.2.2.2

Simplify the left side.

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Step 6.3.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 6.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{5}$

Step 6.3.3.2.2.3

Simplify the right side.

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Step 6.3.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

Step 7

Critical points to evaluate.

$x = - \frac{6}{5}$

Step 8

Evaluate the second derivative at $x = - \frac{6}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{6}{{(5 (- \frac{6}{5}) + 6)}^{\frac{8}{5}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Cancel the common factor of $5$ .

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Step 9.1.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$\frac{6}{{(5 (\frac{- 6}{5}) + 6)}^{\frac{8}{5}}}$

Step 9.1.2

Cancel the common factor.

$\frac{6}{{(5 (\frac{- 6}{5}) + 6)}^{\frac{8}{5}}}$

Step 9.1.3

Rewrite the expression.

$\frac{6}{{(- 6 + 6)}^{\frac{8}{5}}}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Add $- 6$ and $6$ .

$\frac{6}{0^{\frac{8}{5}}}$

Step 9.2.2

Rewrite $0$ as $0^{5}$ .

$\frac{6}{{(0^{5})}^{\frac{8}{5}}}$

Step 9.2.3

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

$\frac{6}{0^{5 (\frac{8}{5})}}$

Step 9.3

Cancel the common factor of $5$ .

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Step 9.3.1

Cancel the common factor.

$\frac{6}{0^{5 (\frac{8}{5})}}$

Step 9.3.2

Rewrite the expression.

$\frac{6}{0^{8}}$

Step 9.4

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

$\frac{6}{0}$

Step 9.5

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

Undefined

Step 10

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

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Step 10.1

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

$(- \infty, - \frac{6}{5}) \cup (- \frac{6}{5}, \infty)$

Step 10.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - \frac{6}{5})$ in the first derivative $- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = - \frac{2}{{(6 + 5 (- 4))}^{\frac{3}{5}}}$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify the denominator.

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Step 10.2.2.1.1

Multiply $5$ by $- 4$ .

$f' (- 4) = - \frac{2}{{(6 - 20)}^{\frac{3}{5}}}$

Step 10.2.2.1.2

Subtract $20$ from $6$ .

$f' (- 4) = - \frac{2}{{(- 14)}^{\frac{3}{5}}}$

Step 10.2.2.2

The final answer is $- \frac{2}{{(- 14)}^{\frac{3}{5}}}$ .

$- \frac{2}{{(- 14)}^{\frac{3}{5}}}$

Step 10.3

Substitute any number, such as $0$ , from the interval $(- \frac{6}{5}, \infty)$ in the first derivative $- \frac{2}{{(6 + 5 x)}^{\frac{3}{5}}}$ to check if the result is negative or positive.

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Step 10.3.1

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

$f' (0) = - \frac{2}{{(6 + 5 (0))}^{\frac{3}{5}}}$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify the denominator.

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Step 10.3.2.1.1

Multiply $5$ by $0$ .

$f' (0) = - \frac{2}{{(6 + 0)}^{\frac{3}{5}}}$

Step 10.3.2.1.2

Add $6$ and $0$ .

$f' (0) = - \frac{2}{6^{\frac{3}{5}}}$

Step 10.3.2.2

The final answer is $- \frac{2}{6^{\frac{3}{5}}}$ .

$- \frac{2}{6^{\frac{3}{5}}}$

Step 10.4

Since the first derivative changed signs from positive to negative around $x = - \frac{6}{5}$ , then $x = - \frac{6}{5}$ is a local maximum.

$x = - \frac{6}{5}$ is a local maximum

Step 11