Calculus Examples

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

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

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 $g (x) = x - 1$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

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

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

Step 1.1.1.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.5.2

Subtract $3$ from $2$ .

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

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

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

Step 1.1.6.2

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

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

Step 1.1.6.3

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

Simplify the expression.

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Step 1.1.10.1

Add $1$ and $0$ .

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

Step 1.1.10.2

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

Since $2 \neq 0$ , 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

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

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

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

Step 4.1.2

Anything raised to $1$ is the base itself.

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

Step 4.2

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

$3 \sqrt[3]{x - 1} = 0$

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.

${(3 \sqrt[3]{x - 1})}^{3} = 0^{3}$

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

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

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

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Step 4.3.2.2.1.1

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

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

Step 4.3.2.2.1.2

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

$27 {({(x - 1)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 4.3.2.2.1.3

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

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Step 4.3.2.2.1.3.1

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

$27 {(x - 1)}^{\frac{1}{3} \cdot 3} = 0^{3}$

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.

$27 {(x - 1)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 1)}^{1} = 0^{3}$

Step 4.3.2.2.1.4

Simplify.

$27 (x - 1) = 0^{3}$

Step 4.3.2.2.1.5

Apply the distributive property.

$27 x + 27 \cdot - 1 = 0^{3}$

Step 4.3.2.2.1.6

Multiply $27$ by $- 1$ .

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

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 $0$ .

$27 x - 27 = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Add $27$ to both sides of the equation.

$27 x = 27$

Step 4.3.3.2

Divide each term in $27 x = 27$ by $27$ and simplify.

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Step 4.3.3.2.1

Divide each term in $27 x = 27$ by $27$ .

$\frac{27 x}{27} = \frac{27}{27}$

Step 4.3.3.2.2

Simplify the left side.

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Step 4.3.3.2.2.1

Cancel the common factor of $27$ .

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Step 4.3.3.2.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{27}{27}$

Step 4.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{27}$

Step 4.3.3.2.3

Simplify the right side.

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Step 4.3.3.2.3.1

Divide $27$ by $27$ .

$x = 1$

Step 5

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

$(- \infty, 1) \cup (1, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

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

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

Subtract $1$ from $0$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Cancel the common factor of $3$ .

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Step 6.2.1.4.1

Cancel the common factor.

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

Step 6.2.1.4.2

Rewrite the expression.

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

Step 6.2.1.5

Evaluate the exponent.

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

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Multiply $3$ by $- 1$ .

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

Step 6.2.2.2

Move the negative in front of the fraction.

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

Step 6.2.3

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

$- \frac{2}{3}$

Step 6.3

At $x = 0$ the derivative is $- \frac{2}{3}$ . Since this is negative, the function is decreasing on $(- \infty, 1)$ .

Decreasing on $(- \infty, 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Subtract $1$ from $2$ .

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

Step 7.2.1.2

One to any power is one.

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

Step 7.2.2

Multiply $3$ by $1$ .

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

Step 7.2.3

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

$\frac{2}{3}$

Step 7.3

At $x = 2$ the derivative is $\frac{2}{3}$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(1, \infty)$

Decreasing on: $(- \infty, 1)$

Step 9