Calculus Examples

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

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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 4$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

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 $n = 2$ .

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

Step 1.1.2.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = x^{2} (2 x + 0) + (x^{2} - 4) \frac{d}{d x} (x^{2})$

Step 1.1.2.4

Add $2 x$ and $0$ .

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

Step 1.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.3.1

Move $x$ .

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

Step 1.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.3.2.1

Raise $x$ to the power of $1$ .

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

Step 1.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = x^{1 + 2} \cdot 2 + (x^{2} - 4) \frac{d}{d x} (x^{2})$

Step 1.1.3.3

Add $1$ and $2$ .

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

Step 1.1.4

Move $2$ to the left of $x^{3}$ .

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

Step 1.1.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.6

Move $2$ to the left of $x^{2} - 4$ .

$f' (x) = 2 x^{3} + 2 \cdot ((x^{2} - 4) x)$

Step 1.1.7

Simplify.

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Step 1.1.7.1

Apply the distributive property.

$f' (x) = 2 x^{3} + (2 x^{2} + 2 \cdot - 4) x$

Step 1.1.7.2

Apply the distributive property.

$f' (x) = 2 x^{3} + 2 x^{2} x + 2 \cdot (- 4 x)$

Step 1.1.7.3

Combine terms.

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Step 1.1.7.3.1

Raise $x$ to the power of $1$ .

$f' (x) = 2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot (- 4 x)$

Step 1.1.7.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = 2 x^{3} + 2 x^{1 + 2} + 2 \cdot (- 4 x)$

Step 1.1.7.3.3

Add $1$ and $2$ .

$f' (x) = 2 x^{3} + 2 x^{3} + 2 \cdot (- 4 x)$

Step 1.1.7.3.4

Multiply $2$ by $- 4$ .

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

Step 1.1.7.3.5

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

$f' (x) = 4 x^{3} - 8 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 8 x$ .

$4 x^{3} - 8 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 8 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 8 x = 0$

Step 2.2

Factor $4 x$ out of $4 x^{3} - 8 x$ .

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Step 2.2.1

Factor $4 x$ out of $4 x^{3}$ .

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

Step 2.2.2

Factor $4 x$ out of $- 8 x$ .

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

Step 2.2.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 2)$ .

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

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 $0$ .

$x = 0$

$x^{2} - 2 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

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

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Step 2.5.1

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

$x^{2} - 2 = 0$

Step 2.5.2

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

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Step 2.5.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 2.5.2.2

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

$x = \pm \sqrt{2}$

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.

$x = \sqrt{2}$

Step 2.5.2.3.2

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

$x = - \sqrt{2}$

Step 2.5.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 2.6

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

$x = 0, \sqrt{2}, - \sqrt{2}$

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 $x^{2} (x^{2} - 4)$ 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 $x$ .

${(0)}^{2} ({(0)}^{2} - 4)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$0 ({(0)}^{2} - 4)$

Step 4.1.2.2

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

$0 (0 - 4)$

Step 4.1.2.3

Subtract $4$ from $0$ .

$0 \cdot - 4$

Step 4.1.2.4

Multiply $0$ by $- 4$ .

$0$

Step 4.2

Evaluate at $x = \sqrt{2}$ .

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Step 4.2.1

Substitute $\sqrt{2}$ for $x$ .

${(\sqrt{2})}^{2} ({(\sqrt{2})}^{2} - 4)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.2.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.4.1

Cancel the common factor.

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

Step 4.2.2.1.4.2

Rewrite the expression.

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

Step 4.2.2.1.5

Evaluate the exponent.

$2 ({(\sqrt{2})}^{2} - 4)$

Step 4.2.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

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

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

Step 4.2.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$2 (2^{\frac{2}{2}} - 4)$

Step 4.2.2.2.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.4.1

Cancel the common factor.

$2 (2^{\frac{2}{2}} - 4)$

Step 4.2.2.2.4.2

Rewrite the expression.

$2 (2^{1} - 4)$

Step 4.2.2.2.5

Evaluate the exponent.

$2 (2 - 4)$

Step 4.2.2.3

Simplify the expression.

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Step 4.2.2.3.1

Subtract $4$ from $2$ .

$2 \cdot - 2$

Step 4.2.2.3.2

Multiply $2$ by $- 2$ .

$- 4$

Step 4.3

Evaluate at $x = - \sqrt{2}$ .

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Step 4.3.1

Substitute $- \sqrt{2}$ for $x$ .

${(- \sqrt{2})}^{2} ({(- \sqrt{2})}^{2} - 4)$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify by cancelling exponent with radical.

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Step 4.3.2.1.1

Apply the product rule to $- \sqrt{2}$ .

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

Step 4.3.2.1.2

Simplify the expression.

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Step 4.3.2.1.2.1

Raise $- 1$ to the power of $2$ .

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

Step 4.3.2.1.2.2

Multiply ${\sqrt{2}}^{2}$ by $1$ .

${\sqrt{2}}^{2} ({(- \sqrt{2})}^{2} - 4)$

Step 4.3.2.1.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.3.2.1.3.1

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

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

Step 4.3.2.1.3.2

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

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

Step 4.3.2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.3.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.3.4.1

Cancel the common factor.

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

Step 4.3.2.1.3.4.2

Rewrite the expression.

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

Step 4.3.2.1.3.5

Evaluate the exponent.

$2 ({(- \sqrt{2})}^{2} - 4)$

Step 4.3.2.2

Simplify each term.

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Step 4.3.2.2.1

Apply the product rule to $- \sqrt{2}$ .

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

Step 4.3.2.2.2

Raise $- 1$ to the power of $2$ .

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

Step 4.3.2.2.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$2 ({\sqrt{2}}^{2} - 4)$

Step 4.3.2.2.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.3.2.2.4.1

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

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

Step 4.3.2.2.4.2

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

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

Step 4.3.2.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$2 (2^{\frac{2}{2}} - 4)$

Step 4.3.2.2.4.4

Cancel the common factor of $2$ .

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Step 4.3.2.2.4.4.1

Cancel the common factor.

$2 (2^{\frac{2}{2}} - 4)$

Step 4.3.2.2.4.4.2

Rewrite the expression.

$2 (2^{1} - 4)$

Step 4.3.2.2.4.5

Evaluate the exponent.

$2 (2 - 4)$

Step 4.3.2.3

Simplify the expression.

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Step 4.3.2.3.1

Subtract $4$ from $2$ .

$2 \cdot - 2$

Step 4.3.2.3.2

Multiply $2$ by $- 2$ .

$- 4$

Step 4.4

List all of the points.

$(0, 0), (\sqrt{2}, - 4), (- \sqrt{2}, - 4)$

Step 5