Calculus Examples

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

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 - x^{3}}$ as ${(2 - x^{3})}^{\frac{1}{3}}$ .

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

Step 1.1.2.2

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 $g (x) = 2 - x^{3}$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $2 - x^{3}$ .

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

Replace all occurrences of $u$ with $2 - x^{3}$ .

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

Step 1.1.2.3

By the Sum Rule, the derivative of $2 - x^{3}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- x^{3}]$ .

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

Step 1.1.2.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Combine the numerators over the common denominator.

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

Step 1.1.2.10

Simplify the numerator.

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Step 1.1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.10.2

Subtract $3$ from $1$ .

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

Step 1.1.2.11

Move the negative in front of the fraction.

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

Step 1.1.2.12

Multiply $3$ by $- 1$ .

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

Step 1.1.2.13

Subtract $3 x^{2}$ from $0$ .

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

Step 1.1.2.14

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

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

Step 1.1.2.15

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

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

Step 1.1.2.16

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

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

Step 1.1.2.17

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

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

Step 1.1.2.18

Factor $3$ out of $- 3 x^{2}$ .

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

Step 1.1.2.19

Cancel the common factors.

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Step 1.1.2.19.1

Factor $3$ out of $3 {(2 - x^{3})}^{\frac{2}{3}}$ .

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

Step 1.1.2.19.2

Cancel the common factor.

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

Step 1.1.2.19.3

Rewrite the expression.

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

Step 1.1.2.20

Move the negative in front of the fraction.

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

Step 1.1.3

Reorder terms.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1$

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.

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

Step 3.2

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

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

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.

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

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

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.2.2

Rewrite the expression.

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

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 $0$ .

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

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Set the $2 - x^{3}$ equal to $0$ .

$2 - x^{3} = 0$

Step 3.3.3.2

Solve for $x$ .

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Step 3.3.3.2.1

Subtract $2$ from both sides of the equation.

$- x^{3} = - 2$

Step 3.3.3.2.2

Divide each term in $- x^{3} = - 2$ by $- 1$ and simplify.

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Step 3.3.3.2.2.1

Divide each term in $- x^{3} = - 2$ by $- 1$ .

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

Step 3.3.3.2.2.2

Simplify the left side.

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Step 3.3.3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.3.3.2.2.2.2

Divide $x^{3}$ by $1$ .

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

Step 3.3.3.2.2.3

Simplify the right side.

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Step 3.3.3.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{3} = 2$

Step 3.3.3.2.3

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

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

Step 4

Evaluate $x + \sqrt[3]{2 - x^{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 1$ .

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Step 4.1.1

Substitute $1$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

One to any power is one.

$1 + \sqrt[3]{2 - 1 \cdot 1}$

Step 4.1.2.1.2

Multiply $- 1$ by $1$ .

$1 + \sqrt[3]{2 - 1}$

Step 4.1.2.1.3

Subtract $1$ from $2$ .

$1 + \sqrt[3]{1}$

Step 4.1.2.1.4

Any root of $1$ is $1$ .

$1 + 1$

Step 4.1.2.2

Add $1$ and $1$ .

$2$

Step 4.2

Evaluate at $x = \sqrt[3]{2}$ .

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Step 4.2.1

Substitute $\sqrt[3]{2}$ for $x$ .

$(\sqrt[3]{2}) + \sqrt[3]{2 - {(\sqrt[3]{2})}^{3}}$

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 ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 4.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

Step 4.2.2.1.1.2

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

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

Step 4.2.2.1.1.3

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

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

Step 4.2.2.1.1.4

Cancel the common factor of $3$ .

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Step 4.2.2.1.1.4.1

Cancel the common factor.

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

Step 4.2.2.1.1.4.2

Rewrite the expression.

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

Step 4.2.2.1.1.5

Evaluate the exponent.

$\sqrt[3]{2} + \sqrt[3]{2 - 1 \cdot 2}$

Step 4.2.2.1.2

Multiply $- 1$ by $2$ .

$\sqrt[3]{2} + \sqrt[3]{2 - 2}$

Step 4.2.2.1.3

Subtract $2$ from $2$ .

$\sqrt[3]{2} + \sqrt[3]{0}$

Step 4.2.2.1.4

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

$\sqrt[3]{2} + \sqrt[3]{0^{3}}$

Step 4.2.2.1.5

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

$\sqrt[3]{2} + 0$

Step 4.2.2.2

Add $\sqrt[3]{2}$ and $0$ .

$\sqrt[3]{2}$

Step 4.3

List all of the points.

$(1, 2), (\sqrt[3]{2}, \sqrt[3]{2})$

Step 5