Calculus Examples

$f (x) = x^{\frac{8}{3}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{3}$ .

$\frac{8}{3} x^{\frac{8}{3} - 1}$

Step 1.2

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

$\frac{8}{3} x^{\frac{8}{3} - 1 \cdot \frac{3}{3}}$

Step 1.3

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

$\frac{8}{3} x^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{8}{3} x^{\frac{8 - 1 \cdot 3}{3}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $3$ .

$\frac{8}{3} x^{\frac{8 - 3}{3}}$

Step 1.5.2

Subtract $3$ from $8$ .

$\frac{8}{3} x^{\frac{5}{3}}$

Step 1.6

Combine $\frac{8}{3}$ and $x^{\frac{5}{3}}$ .

$\frac{8 x^{\frac{5}{3}}}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{8}{3}$ is constant with respect to $x$ , the derivative of $\frac{8 x^{\frac{5}{3}}}{3}$ with respect to $x$ is $\frac{8}{3} \frac{d}{d x} [x^{\frac{5}{3}}]$ .

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

Step 2.2

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

$f'' (x) = \frac{8}{3} \cdot (\frac{5}{3} x^{\frac{5}{3} - 1})$

Step 2.3

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

$f'' (x) = \frac{8}{3} \cdot (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}})$

Step 2.4

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

$f'' (x) = \frac{8}{3} \cdot (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{8}{3} \cdot (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $5$ .

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

Step 2.7

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

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

Step 2.8

Multiply $\frac{8}{3}$ by $\frac{5 x^{\frac{2}{3}}}{3}$ .

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

Step 2.9

Multiply.

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Step 2.9.1

Multiply $5$ by $8$ .

$f'' (x) = \frac{40 x^{\frac{2}{3}}}{3 \cdot 3}$

Step 2.9.2

Multiply $3$ by $3$ .

$f'' (x) = \frac{40 x^{\frac{2}{3}}}{9}$

Step 3

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

$\frac{8 x^{\frac{5}{3}}}{3} = 0$

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 using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{3}$ .

$\frac{8}{3} x^{\frac{8}{3} - 1}$

Step 4.1.2

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

$\frac{8}{3} x^{\frac{8}{3} - 1 \cdot \frac{3}{3}}$

Step 4.1.3

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

$\frac{8}{3} x^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{8}{3} x^{\frac{8 - 1 \cdot 3}{3}}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $- 1$ by $3$ .

$\frac{8}{3} x^{\frac{8 - 3}{3}}$

Step 4.1.5.2

Subtract $3$ from $8$ .

$\frac{8}{3} x^{\frac{5}{3}}$

Step 4.1.6

Combine $\frac{8}{3}$ and $x^{\frac{5}{3}}$ .

$f' (x) = \frac{8 x^{\frac{5}{3}}}{3}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8 x^{\frac{5}{3}}}{3}$ .

$\frac{8 x^{\frac{5}{3}}}{3}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{8 x^{\frac{5}{3}}}{3} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{8 x^{\frac{5}{3}}}{3} = 0$

Step 5.2

Set the numerator equal to zero.

$8 x^{\frac{5}{3}} = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $8 x^{\frac{5}{3}} = 0$ by $8$ and simplify.

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Step 5.3.1.1

Divide each term in $8 x^{\frac{5}{3}} = 0$ by $8$ .

$\frac{8 x^{\frac{5}{3}}}{8} = \frac{0}{8}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor.

$\frac{8 x^{\frac{5}{3}}}{8} = \frac{0}{8}$

Step 5.3.1.2.2

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

$x^{\frac{5}{3}} = \frac{0}{8}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $8$ .

$x^{\frac{5}{3}} = 0$

Step 5.3.2

Raise each side of the equation to the power of $\frac{3}{5}$ to eliminate the fractional exponent on the left side.

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

Step 5.3.3

Simplify the exponent.

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Step 5.3.3.1

Simplify the left side.

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Step 5.3.3.1.1

Simplify ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 5.3.3.1.1.1

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

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Step 5.3.3.1.1.1.1

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

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 0^{\frac{3}{5}}$

Step 5.3.3.1.1.1.2

Cancel the common factor of $5$ .

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Step 5.3.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 0^{\frac{3}{5}}$

Step 5.3.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.1.1.3

Cancel the common factor of $3$ .

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Step 5.3.3.1.1.1.3.1

Cancel the common factor.

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

Step 5.3.3.1.1.1.3.2

Rewrite the expression.

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

Step 5.3.3.1.1.2

Simplify.

$x = 0^{\frac{3}{5}}$

Step 5.3.3.2

Simplify the right side.

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Step 5.3.3.2.1

Simplify $0^{\frac{3}{5}}$ .

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Step 5.3.3.2.1.1

Simplify the expression.

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Step 5.3.3.2.1.1.1

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

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

Step 5.3.3.2.1.1.2

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

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

Step 5.3.3.2.1.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.1.2.1

Cancel the common factor.

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

Step 5.3.3.2.1.2.2

Rewrite the expression.

$x = 0^{3}$

Step 5.3.3.2.1.3

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

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{40 {(0)}^{\frac{2}{3}}}{9}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{40 \cdot {(0^{3})}^{\frac{2}{3}}}{9}$

Step 9.1.2

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

$\frac{40 \cdot 0^{3 (\frac{2}{3})}}{9}$

Step 9.1.3

Cancel the common factor of $3$ .

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Step 9.1.3.1

Cancel the common factor.

$\frac{40 \cdot 0^{3 (\frac{2}{3})}}{9}$

Step 9.1.3.2

Rewrite the expression.

$\frac{40 \cdot 0^{2}}{9}$

Step 9.1.4

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

$\frac{40 \cdot 0}{9}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Multiply $40$ by $0$ .

$\frac{0}{9}$

Step 9.2.2

Divide $0$ by $9$ .

$0$

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.

$(- \infty, 0) \cup (0, \infty)$

Step 10.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{8 x^{\frac{5}{3}}}{3}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{8 {(- 2)}^{\frac{5}{3}}}{3}$

Step 10.2.2

The final answer is $\frac{8 {(- 2)}^{\frac{5}{3}}}{3}$ .

$\frac{8 {(- 2)}^{\frac{5}{3}}}{3}$

Step 10.3

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{8 x^{\frac{5}{3}}}{3}$ to check if the result is negative or positive.

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Step 10.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = \frac{8 {(2)}^{\frac{5}{3}}}{3}$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify the numerator.

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Step 10.3.2.1.1

Rewrite $8$ as $2^{3}$ .

$f' (2) = \frac{2^{3} \cdot 2^{\frac{5}{3}}}{3}$

Step 10.3.2.1.2

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

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

Step 10.3.2.1.3

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

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

Step 10.3.2.1.4

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

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

Step 10.3.2.1.5

Combine the numerators over the common denominator.

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

Step 10.3.2.1.6

Simplify the numerator.

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Step 10.3.2.1.6.1

Multiply $3$ by $3$ .

$f' (2) = \frac{2^{\frac{9 + 5}{3}}}{3}$

Step 10.3.2.1.6.2

Add $9$ and $5$ .

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

Step 10.3.2.2

The final answer is $\frac{2^{\frac{14}{3}}}{3}$ .

$\frac{2^{\frac{14}{3}}}{3}$

Step 10.4

Since the first derivative changed signs from negative to positive around $x = 0$ , then $x = 0$ is a local minimum.

$x = 0$ is a local minimum

Step 11