Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

$f' (x) = 0 + \frac{d}{d x} (- x^{\frac{1}{3}})$

Step 1.1.2

Evaluate $\frac{d}{d x} [- x^{\frac{1}{3}}]$ .

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Step 1.1.2.1

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

$f' (x) = 0 - \frac{d}{d x} x^{\frac{1}{3}}$

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 $n = \frac{1}{3}$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Combine the numerators over the common denominator.

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

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.6.2

Subtract $3$ from $1$ .

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

Step 1.1.2.7

Move the negative in front of the fraction.

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.3

Subtract $\frac{1}{3 x^{\frac{2}{3}}}$ from $0$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

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

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

Step 4.2

Set the denominator in $\frac{1}{3 \sqrt[3]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2}} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

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

${(3 \sqrt[3]{x^{2}})}^{3} = 0^{3}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(3 x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(3 x^{\frac{2}{3}})}^{3}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $3 x^{\frac{2}{3}}$ .

$3^{3} {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.3.2.2.1.2

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

$27 {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 4.3.2.2.1.3.1

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

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{2} = 0^{3}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$27 x^{2} = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Divide each term in $27 x^{2} = 0$ by $27$ and simplify.

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Step 4.3.3.1.1

Divide each term in $27 x^{2} = 0$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 4.3.3.1.2

Simplify the left side.

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Step 4.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 4.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 4.3.3.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{27}$

Step 4.3.3.1.3

Simplify the right side.

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Step 4.3.3.1.3.1

Divide $0$ by $27$ .

$x^{2} = 0$

Step 4.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.3.3.3

Simplify $\pm \sqrt{0}$ .

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Step 4.3.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.3.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

After finding the point that makes the derivative $f' (x) = - \frac{1}{3 x^{\frac{2}{3}}}$ equal to $0$ or undefined, the interval to check where $f (x) = 1 - x^{\frac{1}{3}}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

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

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

Cancel the common factor of $3$ .

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Step 6.2.1.3.1

Cancel the common factor.

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

Step 6.2.1.3.2

Rewrite the expression.

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

Step 6.2.1.4

Raise $- 1$ to the power of $2$ .

$f' (- 1) = - \frac{1}{3 \cdot 1}$

Step 6.2.2

Multiply $3$ by $1$ .

$f' (- 1) = - \frac{1}{3}$

Step 6.2.3

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 6.3

At $x = - 1$ the derivative is $- \frac{1}{3}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

One to any power is one.

$f' (1) = - \frac{1}{3 \cdot 1}$

Step 7.2.2

Multiply $3$ by $1$ .

$f' (1) = - \frac{1}{3}$

Step 7.2.3

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 7.3

At $x = 1$ the derivative is $- \frac{1}{3}$ . Since this is negative, the function is decreasing on $(0, \infty)$ .

Decreasing on $(0, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, 0), (0, \infty)$

Step 9