Calculus Examples

$f (x) = 3 x^{\frac{5}{3}} - 15 x^{\frac{2}{3}}$

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 $3 x^{\frac{5}{3}} - 15 x^{\frac{2}{3}}$ with respect to $x$ is $\frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [- 15 x^{\frac{2}{3}}]$ .

$f' (x) = \frac{d}{d x} (3 x^{\frac{5}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{\frac{5}{3}}]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{5}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{5}{3}}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{\frac{5}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.2

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

$f' (x) = 3 (\frac{5}{3} x^{\frac{5}{3} - 1}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.3

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

$f' (x) = 3 (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.4

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

$f' (x) = 3 (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.5

Combine the numerators over the common denominator.

$f' (x) = 3 (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $3$ .

$f' (x) = 3 (\frac{5}{3} x^{\frac{5 - 3}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.6.2

Subtract $3$ from $5$ .

$f' (x) = 3 (\frac{5}{3} x^{\frac{2}{3}}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.7

Combine $\frac{5}{3}$ and $x^{\frac{2}{3}}$ .

$f' (x) = 3 (\frac{5 x^{\frac{2}{3}}}{3}) + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.8

Combine $3$ and $\frac{5 x^{\frac{2}{3}}}{3}$ .

$f' (x) = \frac{3 (5 x^{\frac{2}{3}})}{3} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.9

Multiply $5$ by $3$ .

$f' (x) = \frac{15 x^{\frac{2}{3}}}{3} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.10

Factor $3$ out of $15 x^{\frac{2}{3}}$ .

$f' (x) = \frac{3 (5 x^{\frac{2}{3}})}{3} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.11

Cancel the common factors.

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Step 1.1.2.11.1

Factor $3$ out of $3$ .

$f' (x) = \frac{3 (5 x^{\frac{2}{3}})}{3 (1)} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.11.2

Cancel the common factor.

$f' (x) = \frac{3 (5 x^{\frac{2}{3}})}{3 \cdot 1} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.11.3

Rewrite the expression.

$f' (x) = \frac{5 x^{\frac{2}{3}}}{1} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.2.11.4

Divide $5 x^{\frac{2}{3}}$ by $1$ .

$f' (x) = 5 x^{\frac{2}{3}} + \frac{d}{d x} (- 15 x^{\frac{2}{3}})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 15 x^{\frac{2}{3}}]$ .

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Step 1.1.3.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x^{\frac{2}{3}}$ with respect to $x$ is $- 15 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$f' (x) = 5 x^{\frac{2}{3}} - 15 \frac{d}{d x} x^{\frac{2}{3}}$

Step 1.1.3.2

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

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 1.1.3.3

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

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.3.4

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

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $3$ .

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 1.1.3.6.2

Subtract $3$ from $2$ .

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 1.1.3.7

Move the negative in front of the fraction.

$f' (x) = 5 x^{\frac{2}{3}} - 15 (\frac{2}{3} x^{- \frac{1}{3}})$

Step 1.1.3.8

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

$f' (x) = 5 x^{\frac{2}{3}} - 15 \frac{2 x^{- \frac{1}{3}}}{3}$

Step 1.1.3.9

Combine $- 15$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$f' (x) = 5 x^{\frac{2}{3}} + \frac{- 15 (2 x^{- \frac{1}{3}})}{3}$

Step 1.1.3.10

Multiply $2$ by $- 15$ .

$f' (x) = 5 x^{\frac{2}{3}} + \frac{- 30 x^{- \frac{1}{3}}}{3}$

Step 1.1.3.11

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

$f' (x) = 5 x^{\frac{2}{3}} + \frac{- 30}{3 x^{\frac{1}{3}}}$

Step 1.1.3.12

Factor $3$ out of $- 30$ .

$f' (x) = 5 x^{\frac{2}{3}} + \frac{3 \cdot - 10}{3 x^{\frac{1}{3}}}$

Step 1.1.3.13

Cancel the common factors.

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Step 1.1.3.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$f' (x) = 5 x^{\frac{2}{3}} + \frac{3 \cdot - 10}{3 (x^{\frac{1}{3}})}$

Step 1.1.3.13.2

Cancel the common factor.

$f' (x) = 5 x^{\frac{2}{3}} + \frac{3 \cdot - 10}{3 x^{\frac{1}{3}}}$

Step 1.1.3.13.3

Rewrite the expression.

$f' (x) = 5 x^{\frac{2}{3}} + \frac{- 10}{x^{\frac{1}{3}}}$

Step 1.1.3.14

Move the negative in front of the fraction.

$f' (x) = 5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}}$ .

$5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{\frac{1}{3}}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{\frac{1}{3}}$

Step 2.3

Multiply each term in $5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}} = 0$ by $x^{\frac{1}{3}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $5 x^{\frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}} = 0$ by $x^{\frac{1}{3}}$ .

$5 x^{\frac{2}{3}} x^{\frac{1}{3}} - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{\frac{1}{3}}$ .

$5 (x^{\frac{1}{3}} x^{\frac{2}{3}}) - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.1.2

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

$5 x^{\frac{1}{3} + \frac{2}{3}} - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.1.3

Combine the numerators over the common denominator.

$5 x^{\frac{1 + 2}{3}} - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.1.4

Add $1$ and $2$ .

$5 x^{\frac{3}{3}} - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.1.5

Divide $3$ by $3$ .

$5 x^{1} - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.2

Simplify $5 x^{1}$ .

$5 x - \frac{10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.3

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 2.3.2.1.3.1

Move the leading negative in $- \frac{10}{x^{\frac{1}{3}}}$ into the numerator.

$5 x + \frac{- 10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.3.2

Cancel the common factor.

$5 x + \frac{- 10}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 0 x^{\frac{1}{3}}$

Step 2.3.2.1.3.3

Rewrite the expression.

$5 x - 10 = 0 x^{\frac{1}{3}}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{\frac{1}{3}}$ .

$5 x - 10 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $10$ to both sides of the equation.

$5 x = 10$

Step 2.4.2

Divide each term in $5 x = 10$ by $5$ and simplify.

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Step 2.4.2.1

Divide each term in $5 x = 10$ by $5$ .

$\frac{5 x}{5} = \frac{10}{5}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{10}{5}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{5}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $10$ by $5$ .

$x = 2$

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.

$5 \sqrt[3]{x^{2}} - \frac{10}{x^{\frac{1}{3}}}$

Step 3.1.2

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

$5 \sqrt[3]{x^{2}} - \frac{10}{\sqrt[3]{x^{1}}}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$5 \sqrt[3]{x^{2}} - \frac{10}{\sqrt[3]{x}}$

Step 3.2

Set the denominator in $\frac{10}{\sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x} = 0$

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.

${\sqrt[3]{x}}^{3} = 0^{3}$

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

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

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Step 3.3.2.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 3.3.2.2.1.2

Simplify.

$x = 0^{3}$

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 $0$ .

$x = 0$

Step 4

Evaluate $3 x^{\frac{5}{3}} - 15 x^{\frac{2}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 2$ .

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Step 4.1.1

Substitute $2$ for $x$ .

$3 {(2)}^{\frac{5}{3}} - 15 {(2)}^{\frac{2}{3}}$

Step 4.1.2

Remove parentheses.

$3 \cdot 2^{\frac{5}{3}} - 15 \cdot 2^{\frac{2}{3}}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$3 {(0)}^{\frac{5}{3}} - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

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

$3 {(0^{3})}^{\frac{5}{3}} - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.2

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

$3 \cdot 0^{3 (\frac{5}{3})} - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.3

Cancel the common factor of $3$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

$3 \cdot 0^{3 (\frac{5}{3})} - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.3.2

Rewrite the expression.

$3 \cdot 0^{5} - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.4

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

$3 \cdot 0 - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.5

Multiply $3$ by $0$ .

$0 - 15 {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.6

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

$0 - 15 {(0^{3})}^{\frac{2}{3}}$

Step 4.2.2.1.7

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

$0 - 15 \cdot 0^{3 (\frac{2}{3})}$

Step 4.2.2.1.8

Cancel the common factor of $3$ .

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Step 4.2.2.1.8.1

Cancel the common factor.

$0 - 15 \cdot 0^{3 (\frac{2}{3})}$

Step 4.2.2.1.8.2

Rewrite the expression.

$0 - 15 \cdot 0^{2}$

Step 4.2.2.1.9

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

$0 - 15 \cdot 0$

Step 4.2.2.1.10

Multiply $- 15$ by $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(2, 3 \cdot 2^{\frac{5}{3}} - 15 \cdot 2^{\frac{2}{3}}), (0, 0)$

Step 5