Calculus Examples

$f (x) = x^{5} - 10 x^{3} - 8$

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 $x^{5} - 10 x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 10 x^{3}] + \frac{d}{d x} [- 8]$ .

$f' (x) = \frac{d}{d x} (x^{5}) + \frac{d}{d x} (- 10 x^{3}) + \frac{d}{d x} (- 8)$

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 $n = 5$ .

$f' (x) = 5 x^{4} + \frac{d}{d x} (- 10 x^{3}) + \frac{d}{d x} (- 8)$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 10 x^{3}]$ .

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Step 1.1.2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{3}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 5 x^{4} - 10 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 8)$

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 $n = 3$ .

$f' (x) = 5 x^{4} - 10 (3 x^{2}) + \frac{d}{d x} (- 8)$

Step 1.1.2.3

Multiply $3$ by $- 10$ .

$f' (x) = 5 x^{4} - 30 x^{2} + \frac{d}{d x} (- 8)$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

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

$f' (x) = 5 x^{4} - 30 x^{2} + 0$

Step 1.1.3.2

Add $5 x^{4} - 30 x^{2}$ and $0$ .

$f' (x) = 5 x^{4} - 30 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{4} - 30 x^{2}$ .

$5 x^{4} - 30 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $5 x^{4} - 30 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$5 x^{4} - 30 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$5 {(x^{2})}^{2} - 30 x^{2} = 0$

Step 2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$5 u^{2} - 30 u = 0$

Step 2.2.3

Factor $5 u$ out of $5 u^{2} - 30 u$ .

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Step 2.2.3.1

Factor $5 u$ out of $5 u^{2}$ .

$5 u (u) - 30 u = 0$

Step 2.2.3.2

Factor $5 u$ out of $- 30 u$ .

$5 u (u) + 5 u (- 6) = 0$

Step 2.2.3.3

Factor $5 u$ out of $5 u (u) + 5 u (- 6)$ .

$5 u (u - 6) = 0$

Step 2.2.4

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

$5 x^{2} (x^{2} - 6) = 0$

Step 2.3

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

$x^{2} = 0$

$x^{2} - 6 = 0$

Step 2.4

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

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Step 2.4.1

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

$x^{2} = 0$

Step 2.4.2

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

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 2.5.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 2.5.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 2.5.2.2

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

$x = \pm \sqrt{6}$

Step 2.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{6}$

Step 2.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{6}$

Step 2.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{6}, - \sqrt{6}$

Step 2.6

The final solution is all the values that make $5 x^{2} (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

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 $x^{5} - 10 x^{3} - 8$ 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 $x$ .

${(0)}^{5} - 10 {(0)}^{3} - 8$

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

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

$0 - 10 {(0)}^{3} - 8$

Step 4.1.2.1.2

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

$0 - 10 \cdot 0 - 8$

Step 4.1.2.1.3

Multiply $- 10$ by $0$ .

$0 + 0 - 8$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$0 - 8$

Step 4.1.2.2.2

Subtract $8$ from $0$ .

$- 8$

Step 4.2

Evaluate at $x = \sqrt{6}$ .

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Step 4.2.1

Substitute $\sqrt{6}$ for $x$ .

${(\sqrt{6})}^{5} - 10 {(\sqrt{6})}^{3} - 8$

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 ${\sqrt{6}}^{5}$ as $\sqrt{6^{5}}$ .

$\sqrt{6^{5}} - 10 {(\sqrt{6})}^{3} - 8$

Step 4.2.2.1.2

Raise $6$ to the power of $5$ .

$\sqrt{7776} - 10 {(\sqrt{6})}^{3} - 8$

Step 4.2.2.1.3

Rewrite $7776$ as $36^{2} \cdot 6$ .

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Step 4.2.2.1.3.1

Factor $1296$ out of $7776$ .

$\sqrt{1296 (6)} - 10 {(\sqrt{6})}^{3} - 8$

Step 4.2.2.1.3.2

Rewrite $1296$ as $36^{2}$ .

$\sqrt{36^{2} \cdot 6} - 10 {(\sqrt{6})}^{3} - 8$

Step 4.2.2.1.4

Pull terms out from under the radical.

$36 \sqrt{6} - 10 {(\sqrt{6})}^{3} - 8$

Step 4.2.2.1.5

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$36 \sqrt{6} - 10 \sqrt{6^{3}} - 8$

Step 4.2.2.1.6

Raise $6$ to the power of $3$ .

$36 \sqrt{6} - 10 \sqrt{216} - 8$

Step 4.2.2.1.7

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 4.2.2.1.7.1

Factor $36$ out of $216$ .

$36 \sqrt{6} - 10 \sqrt{36 (6)} - 8$

Step 4.2.2.1.7.2

Rewrite $36$ as $6^{2}$ .

$36 \sqrt{6} - 10 \sqrt{6^{2} \cdot 6} - 8$

Step 4.2.2.1.8

Pull terms out from under the radical.

$36 \sqrt{6} - 10 (6 \sqrt{6}) - 8$

Step 4.2.2.1.9

Multiply $6$ by $- 10$ .

$36 \sqrt{6} - 60 \sqrt{6} - 8$

Step 4.2.2.2

Subtract $60 \sqrt{6}$ from $36 \sqrt{6}$ .

$- 24 \sqrt{6} - 8$

Step 4.3

Evaluate at $x = - \sqrt{6}$ .

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Step 4.3.1

Substitute $- \sqrt{6}$ for $x$ .

${(- \sqrt{6})}^{5} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Apply the product rule to $- \sqrt{6}$ .

${(- 1)}^{5} {\sqrt{6}}^{5} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.2

Raise $- 1$ to the power of $5$ .

$- {\sqrt{6}}^{5} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.3

Rewrite ${\sqrt{6}}^{5}$ as $\sqrt{6^{5}}$ .

$- \sqrt{6^{5}} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.4

Raise $6$ to the power of $5$ .

$- \sqrt{7776} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.5

Rewrite $7776$ as $36^{2} \cdot 6$ .

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Step 4.3.2.1.5.1

Factor $1296$ out of $7776$ .

$- \sqrt{1296 (6)} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.5.2

Rewrite $1296$ as $36^{2}$ .

$- \sqrt{36^{2} \cdot 6} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.6

Pull terms out from under the radical.

$- (36 \sqrt{6}) - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.7

Multiply $36$ by $- 1$ .

$- 36 \sqrt{6} - 10 {(- \sqrt{6})}^{3} - 8$

Step 4.3.2.1.8

Apply the product rule to $- \sqrt{6}$ .

$- 36 \sqrt{6} - 10 ({(- 1)}^{3} {\sqrt{6}}^{3}) - 8$

Step 4.3.2.1.9

Raise $- 1$ to the power of $3$ .

$- 36 \sqrt{6} - 10 (- {\sqrt{6}}^{3}) - 8$

Step 4.3.2.1.10

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$- 36 \sqrt{6} - 10 (- \sqrt{6^{3}}) - 8$

Step 4.3.2.1.11

Raise $6$ to the power of $3$ .

$- 36 \sqrt{6} - 10 (- \sqrt{216}) - 8$

Step 4.3.2.1.12

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 4.3.2.1.12.1

Factor $36$ out of $216$ .

$- 36 \sqrt{6} - 10 (- \sqrt{36 (6)}) - 8$

Step 4.3.2.1.12.2

Rewrite $36$ as $6^{2}$ .

$- 36 \sqrt{6} - 10 (- \sqrt{6^{2} \cdot 6}) - 8$

Step 4.3.2.1.13

Pull terms out from under the radical.

$- 36 \sqrt{6} - 10 (- (6 \sqrt{6})) - 8$

Step 4.3.2.1.14

Multiply $6$ by $- 1$ .

$- 36 \sqrt{6} - 10 (- 6 \sqrt{6}) - 8$

Step 4.3.2.1.15

Multiply $- 6$ by $- 10$ .

$- 36 \sqrt{6} + 60 \sqrt{6} - 8$

Step 4.3.2.2

Add $- 36 \sqrt{6}$ and $60 \sqrt{6}$ .

$24 \sqrt{6} - 8$

Step 4.4

List all of the points.

$(0, - 8), (\sqrt{6}, - 24 \sqrt{6} - 8), (- \sqrt{6}, 24 \sqrt{6} - 8)$

Step 5