Calculus Examples

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

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 using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x - 3$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Replace all occurrences of $u$ with $x - 3$ .

$\frac{1}{3} {(x - 3)}^{\frac{1}{3} - 1} \frac{d}{d x} [x - 3]$

Step 1.1.2

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

$\frac{1}{3} {(x - 3)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 3]$

Step 1.1.3

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

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

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{1}{3} {(x - 3)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 3]$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(x - 3)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x - 3]$

Step 1.1.5.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(x - 3)}^{\frac{- 2}{3}} \frac{d}{d x} [x - 3]$

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

$\frac{1}{3} {(x - 3)}^{- \frac{2}{3}} \frac{d}{d x} [x - 3]$

Step 1.1.6.2

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

$\frac{{(x - 3)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x - 3]$

Step 1.1.6.3

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

$\frac{1}{3 {(x - 3)}^{\frac{2}{3}}} \frac{d}{d x} [x - 3]$

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 1.1.10

Simplify the expression.

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Step 1.1.10.1

Add $1$ and $0$ .

$\frac{1}{3 {(x - 3)}^{\frac{2}{3}}} \cdot 1$

Step 1.1.10.2

Multiply $\frac{1}{3 {(x - 3)}^{\frac{2}{3}}}$ by $1$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

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

Find the values where the derivative is undefined.

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Step 3.1

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

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

Step 3.2

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

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

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.

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

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 $\sqrt[3]{{(x - 3)}^{2}}$ as ${(x - 3)}^{\frac{2}{3}}$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

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

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

Step 3.3.2.2.1.3

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

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Step 3.3.2.2.1.3.1

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

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

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.

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

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

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

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 $0$ .

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

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Divide each term in $27 {(x - 3)}^{2} = 0$ by $27$ and simplify.

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Step 3.3.3.1.1

Divide each term in $27 {(x - 3)}^{2} = 0$ by $27$ .

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

Step 3.3.3.1.2

Simplify the left side.

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Step 3.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 3.3.3.1.2.1.1

Cancel the common factor.

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

Step 3.3.3.1.2.1.2

Divide ${(x - 3)}^{2}$ by $1$ .

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

Step 3.3.3.1.3

Simplify the right side.

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Step 3.3.3.1.3.1

Divide $0$ by $27$ .

${(x - 3)}^{2} = 0$

Step 3.3.3.2

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.3.3.3

Add $3$ to both sides of the equation.

$x = 3$

Step 4

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

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

${((3) - 3)}^{\frac{1}{3}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Subtract $3$ from $3$ .

$0^{\frac{1}{3}}$

Step 4.1.2.1.2

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

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

Step 4.1.2.1.3

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

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

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

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

Step 4.1.2.2.2

Rewrite the expression.

$0^{1}$

Step 4.1.2.3

Evaluate the exponent.

$0$

Step 4.2

List all of the points.

$(3, 0)$

Step 5