Calculus Examples

${(x^{3} - 8)}^{4}$

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 $f (x) = x^{4}$ and $g (x) = x^{3} - 8$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x^{3} - 8$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [x^{3} - 8]$

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 $n = 4$ .

$4 u^{3} \frac{d}{d x} [x^{3} - 8]$

Step 1.1.1.3

Replace all occurrences of $u$ with $x^{3} - 8$ .

$4 {(x^{3} - 8)}^{3} \frac{d}{d x} [x^{3} - 8]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

$4 {(x^{3} - 8)}^{3} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8])$

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 $n = 3$ .

$4 {(x^{3} - 8)}^{3} (3 x^{2} + \frac{d}{d x} [- 8])$

Step 1.1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$4 {(x^{3} - 8)}^{3} (3 x^{2} + 0)$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $3 x^{2}$ and $0$ .

$4 {(x^{3} - 8)}^{3} (3 x^{2})$

Step 1.1.2.4.2

Multiply $3$ by $4$ .

$12 {(x^{3} - 8)}^{3} x^{2}$

Step 1.1.2.4.3

Reorder the factors of $12 {(x^{3} - 8)}^{3} x^{2}$ .

$f' (x) = 12 x^{2} {(x^{3} - 8)}^{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} {(x^{3} - 8)}^{3}$ .

$12 x^{2} {(x^{3} - 8)}^{3}$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} {(x^{3} - 8)}^{3} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} {(x^{3} - 8)}^{3} = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

${(x^{3} - 8)}^{3} = 0$

Step 2.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.3.2.1

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

$x = \pm \sqrt{0}$

Step 2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

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

$x = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

Set ${(x^{3} - 8)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x^{3} - 8)}^{3}$ equal to $0$ .

${(x^{3} - 8)}^{3} = 0$

Step 2.4.2

Solve ${(x^{3} - 8)}^{3} = 0$ for $x$ .

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Step 2.4.2.1

Factor the left side of the equation.

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Step 2.4.2.1.1

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

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

Step 2.4.2.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 2.4.2.1.3

Simplify.

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Step 2.4.2.1.3.1

Move $2$ to the left of $x$ .

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

Step 2.4.2.1.3.2

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

${((x - 2) (x^{2} + 2 x + 4))}^{3} = 0$

Step 2.4.2.1.4

Apply the product rule to $(x - 2) (x^{2} + 2 x + 4)$ .

${(x - 2)}^{3} {(x^{2} + 2 x + 4)}^{3} = 0$

Step 2.4.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

${(x^{2} + 2 x + 4)}^{3} = 0$

Step 2.4.2.3

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.2.3.1

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

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

Step 2.4.2.3.2

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

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Step 2.4.2.3.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.4.2.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.4.2.4

Set ${(x^{2} + 2 x + 4)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.2.4.1

Set ${(x^{2} + 2 x + 4)}^{3}$ equal to $0$ .

${(x^{2} + 2 x + 4)}^{3} = 0$

Step 2.4.2.4.2

Solve ${(x^{2} + 2 x + 4)}^{3} = 0$ for $x$ .

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Step 2.4.2.4.2.1

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

$x^{2} + 2 x + 4 = \sqrt[3]{0}$

Step 2.4.2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.4.2.4.2.2.1

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

$x^{2} + 2 x + 4 = \sqrt[3]{0^{3}}$

Step 2.4.2.4.2.2.2

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

$x^{2} + 2 x + 4 = 0$

Step 2.4.2.4.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.2.4.2.4

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 2.4.2.4.2.5

Simplify.

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Step 2.4.2.4.2.5.1

Simplify the numerator.

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Step 2.4.2.4.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.4.2.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.4.2.4.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.4.2.4.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.2.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.4.2.4.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.4.2.4.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.2.4.2.6.1

Simplify the numerator.

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Step 2.4.2.4.2.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.4.2.4.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.4.2.4.2.6.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.4.2.4.2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.2.4.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.4.2.4.2.6.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.4.2.4.2.6.4

Change the $\pm$ to $+$ .

$x = - 1 + i \sqrt{3}$

Step 2.4.2.4.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.2.4.2.7.1

Simplify the numerator.

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Step 2.4.2.4.2.7.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.4.2.4.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.4.2.4.2.7.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.4.2.4.2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.2.4.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.4.2.4.2.7.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.4.2.4.2.7.4

Change the $\pm$ to $-$ .

$x = - 1 - i \sqrt{3}$

Step 2.4.2.4.2.8

The final answer is the combination of both solutions.

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 2.4.2.5

The final solution is all the values that make ${(x - 2)}^{3} {(x^{2} + 2 x + 4)}^{3} = 0$ true.

$x = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 2.5

The final solution is all the values that make $12 x^{2} {(x^{3} - 8)}^{3} = 0$ true.

$x = 0, 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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 4

Evaluate ${(x^{3} - 8)}^{4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${({(0)}^{3} - 8)}^{4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

${(0 - 8)}^{4}$

Step 4.1.2.2

Subtract $8$ from $0$ .

${(- 8)}^{4}$

Step 4.1.2.3

Raise $- 8$ to the power of $4$ .

$4096$

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

${({(2)}^{3} - 8)}^{4}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $2$ to the power of $3$ .

${(8 - 8)}^{4}$

Step 4.2.2.2

Subtract $8$ from $8$ .

$0^{4}$

Step 4.2.2.3

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

$0$

Step 4.3

List all of the points.

$(0, 4096), (2, 0)$

Step 5