Calculus Examples

$y = | x^{2} - 9 |$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (| x^{2} - 9 |)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 9$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 9$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 9]$

Step 3.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - 9]$

Step 3.1.3

Replace all occurrences of $u$ with $x^{2} - 9$ .

$\frac{x^{2} - 9}{| x^{2} - 9 |} \frac{d}{d x} [x^{2} - 9]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$ .

$\frac{x^{2} - 9}{| x^{2} - 9 |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9])$

Step 3.2.2

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

$\frac{x^{2} - 9}{| x^{2} - 9 |} (2 x + \frac{d}{d x} [- 9])$

Step 3.2.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{x^{2} - 9}{| x^{2} - 9 |} (2 x + 0)$

Step 3.2.4

Combine fractions.

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Step 3.2.4.1

Add $2 x$ and $0$ .

$\frac{x^{2} - 9}{| x^{2} - 9 |} (2 x)$

Step 3.2.4.2

Combine $2$ and $\frac{x^{2} - 9}{| x^{2} - 9 |}$ .

$\frac{2 (x^{2} - 9)}{| x^{2} - 9 |} x$

Step 3.2.4.3

Combine $\frac{2 (x^{2} - 9)}{| x^{2} - 9 |}$ and $x$ .

$\frac{2 (x^{2} - 9) x}{| x^{2} - 9 |}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot - 9) x}{| x^{2} - 9 |}$

Step 3.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.3.1.2

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

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Step 3.3.3.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.3.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.3.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.3.2

Multiply $2$ by $- 9$ .

$\frac{2 x^{3} - 18 x}{| x^{2} - 9 |}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2 x^{3} - 18 x}{| x^{2} - 9 |}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 x^{3} - 18 x}{| x^{2} - 9 |}$

Step 6

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 6.1

Set the numerator equal to zero.

$2 x^{3} - 18 x = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Factor the left side of the equation.

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Step 6.2.1.1

Factor $2 x$ out of $2 x^{3} - 18 x$ .

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Step 6.2.1.1.1

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

$2 x (x^{2}) - 18 x = 0$

Step 6.2.1.1.2

Factor $2 x$ out of $- 18 x$ .

$2 x (x^{2}) + 2 x (- 9) = 0$

Step 6.2.1.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 9)$ .

$2 x (x^{2} - 9) = 0$

Step 6.2.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 6.2.1.3

Factor.

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Step 6.2.1.3.1

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

$2 x ((x + 3) (x - 3)) = 0$

Step 6.2.1.3.2

Remove unnecessary parentheses.

$2 x (x + 3) (x - 3) = 0$

Step 6.2.2

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

$x = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 6.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.2.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 6.2.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 6.2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 6.2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6.2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.2.6

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

$x = 0, - 3, 3$

Step 6.3

Exclude the solutions that do not make $\frac{2 x^{3} - 18 x}{| x^{2} - 9 |} = 0$ true.

$x = 0$

Step 7

Simplify $| {(0)}^{2} - 9 |$ .

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Step 7.1

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

$y = | 0 - 9 |$

Step 7.2

Subtract $9$ from $0$ .

$y = | - 9 |$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $- 9$ and $0$ is $9$ .

$y = 9$

Step 8

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 9)$

Step 9