Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 1.1.2

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

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Step 1.1.2.1

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

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Step 1.1.2.2.1

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 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 $n = \frac{2}{3}$ .

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

Step 1.1.2.2.3

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 1.1.2.3

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 1.1.2.4

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 1.1.2.5

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 1.1.2.6

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 1.1.2.7

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 1.1.2.8

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 1.1.2.9

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 1.1.2.10

Simplify the numerator.

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Step 1.1.2.10.1

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 1.1.2.10.2

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 1.1.2.11

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 1.1.2.12

Multiply $- 9$ by $1$ .

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

Step 1.1.2.13

Subtract $9$ from $0$ .

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

Step 1.1.2.14

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 1.1.2.15

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 1.1.2.16

Multiply $- 9$ by $2$ .

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

Step 1.1.2.17

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 1.1.2.18

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 1.1.2.19

Cancel the common factors.

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Step 1.1.2.19.1

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 1.1.2.19.2

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 1.1.2.19.3

Rewrite the expression.

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

Step 1.1.2.20

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 1.1.2.21

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 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

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 1.1.3.2

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

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

Step 1.2

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 2

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 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$6 = 0$

Step 2.3

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

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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 3.1.2

Anything raised to $1$ is the base itself.

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

Step 3.2

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 3.3

Solve for $x$ .

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Step 3.3.1

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 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

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 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

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

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

Step 3.3.2.2.1.3

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

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Step 3.3.2.2.1.3.1

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 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 3.3.2.2.1.4

Simplify.

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

Step 3.3.2.2.1.5

Apply the distributive property.

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

Step 3.3.2.2.1.6

Multiply.

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Step 3.3.2.2.1.6.1

Multiply $343$ by $5$ .

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

Step 3.3.2.2.1.6.2

Multiply $- 9$ by $343$ .

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

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$1715 - 3087 x = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $1715$ from both sides of the equation.

$- 3087 x = - 1715$

Step 3.3.3.2

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

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $- 3087$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

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

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Step 3.3.3.2.3.1.1

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

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

Step 3.3.3.2.3.1.2

Cancel the common factors.

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Step 3.3.3.2.3.1.2.1

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

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

Step 3.3.3.2.3.1.2.2

Cancel the common factor.

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

Step 3.3.3.2.3.1.2.3

Rewrite the expression.

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

Step 4

Evaluate $\frac{{(5 - 9 x)}^{\frac{2}{3}}}{7} + 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{5}{9}$ .

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Step 4.1.1

Substitute $\frac{5}{9}$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Simplify the numerator.

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Step 4.1.2.1.1.1

Simplify each term.

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Step 4.1.2.1.1.1.1

Cancel the common factor of $9$ .

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Step 4.1.2.1.1.1.1.1

Factor $9$ out of $- 9$ .

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

Step 4.1.2.1.1.1.1.2

Cancel the common factor.

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

Step 4.1.2.1.1.1.1.3

Rewrite the expression.

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

Step 4.1.2.1.1.1.2

Multiply $- 1$ by $5$ .

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

Step 4.1.2.1.1.2

Subtract $5$ from $5$ .

$\frac{0^{\frac{2}{3}}}{7} + 1$

Step 4.1.2.1.1.3

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

$\frac{{(0^{3})}^{\frac{2}{3}}}{7} + 1$

Step 4.1.2.1.1.4

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

$\frac{0^{3 (\frac{2}{3})}}{7} + 1$

Step 4.1.2.1.1.5

Cancel the common factor of $3$ .

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Step 4.1.2.1.1.5.1

Cancel the common factor.

$\frac{0^{3 (\frac{2}{3})}}{7} + 1$

Step 4.1.2.1.1.5.2

Rewrite the expression.

$\frac{0^{2}}{7} + 1$

Step 4.1.2.1.1.6

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

$\frac{0}{7} + 1$

Step 4.1.2.1.2

Divide $0$ by $7$ .

$0 + 1$

Step 4.1.2.2

Add $0$ and $1$ .

$1$

Step 4.2

List all of the points.

$(\frac{5}{9}, 1)$

Step 5