Algebra Examples

$f (x) = | x - a | - b$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $| x - a | - b$ with respect to $x$ is $\frac{d}{d x} [| x - a |] + \frac{d}{d x} [- b]$ .

$\frac{d}{d x} [| x - a |] + \frac{d}{d x} [- b]$

Step 1.2

Evaluate $\frac{d}{d x} [| x - a |]$ .

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Step 1.2.1

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 $g (x) = x - a$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $x - a$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 1.2.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 1.2.1.3

Replace all occurrences of $u$ with $x - a$ .

$\frac{x - a}{| x - a |} \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 1.2.2

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

$\frac{x - a}{| x - a |} (\frac{d}{d x} [x] + \frac{d}{d x} [- a]) + \frac{d}{d x} [- b]$

Step 1.2.3

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

$\frac{x - a}{| x - a |} (1 + \frac{d}{d x} [- a]) + \frac{d}{d x} [- b]$

Step 1.2.4

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

$\frac{x - a}{| x - a |} (1 + 0) + \frac{d}{d x} [- b]$

Step 1.2.5

Add $1$ and $0$ .

$\frac{x - a}{| x - a |} \cdot 1 + \frac{d}{d x} [- b]$

Step 1.2.6

Multiply $\frac{x - a}{| x - a |}$ by $1$ .

$\frac{x - a}{| x - a |} + \frac{d}{d x} [- b]$

Step 1.3

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

$\frac{x - a}{| x - a |} + 0$

Step 1.4

Simplify.

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Step 1.4.1

Add $\frac{x - a}{| x - a |}$ and $0$ .

$\frac{x - a}{| x - a |}$

Step 1.4.2

Reorder terms.

$\frac{- a + x}{| x - a |}$

Step 1.4.3

Factor $- 1$ out of $- a$ .

$\frac{- (a) + x}{| x - a |}$

Step 1.4.4

Factor $- 1$ out of $x$ .

$\frac{- (a) - 1 (- x)}{| x - a |}$

Step 1.4.5

Factor $- 1$ out of $- (a) - 1 (- x)$ .

$\frac{- (a - x)}{| x - a |}$

Step 1.4.6

Rewrite $- (a - x)$ as $- 1 (a - x)$ .

$\frac{- 1 (a - x)}{| x - a |}$

Step 1.4.7

Move the negative in front of the fraction.

$- \frac{a - x}{| x - a |}$

Step 2

Find the second derivative of the function.

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Step 2.1

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 $f (x) = - 1$ and $g (x) = \frac{a - x}{| x - a |}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{a - x}{| x - a |} + \frac{a - x}{| x - a |} \cdot \frac{d}{d x} (- 1)$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = a - x$ and $g (x) = | x - a |$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

$f'' (x) = - \frac{| x - a | (0 + \frac{d}{d x} (- x)) - (a - x) \frac{d}{d x} | x - a |}{{| x - a |}^{2}} + \frac{a - x}{| x - a |} \cdot \frac{d}{d x} (- 1)$

Step 2.3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Multiply $- 1$ by $1$ .

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

Step 2.3.6.2

Move $- 1$ to the left of $| x - a |$ .

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

Step 2.3.6.3

Rewrite $- 1 | x - a |$ as $- | x - a |$ .

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

Step 2.4

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 $g (x) = x - a$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x - a$ .

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u$ with $x - a$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

$f'' (x) = - \frac{- | x - a | - (a - x) (\frac{x - a}{| x - a |} \cdot (1 + 0))}{{| x - a |}^{2}} + \frac{a - x}{| x - a |} \cdot \frac{d}{d x} (- 1)$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

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

Step 2.5.4.2

Multiply $\frac{x - a}{| x - a |}$ by $1$ .

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

Step 2.5.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{- | x - a | - (a - x) \frac{x - a}{| x - a |}}{{| x - a |}^{2}} + \frac{a - x}{| x - a |} \cdot 0$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Multiply $\frac{a - x}{| x - a |}$ by $0$ .

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

Step 2.5.6.2

Add $- \frac{- | x - a | - (a - x) \frac{x - a}{| x - a |}}{{| x - a |}^{2}}$ and $0$ .

$f'' (x) = - \frac{- | x - a | - (a - x) \frac{x - a}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$f'' (x) = - \frac{- | x - a | + (- a + x) (\frac{x - a}{| x - a |})}{{| x - a |}^{2}}$

Step 2.6.2

Simplify the numerator.

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Step 2.6.2.1

Simplify each term.

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Step 2.6.2.1.1

Multiply $- - x$ .

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Step 2.6.2.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.2.1.1.2

Multiply $x$ by $1$ .

$f'' (x) = - \frac{- | x - a | + (- a + x) (\frac{x - a}{| x - a |})}{{| x - a |}^{2}}$

Step 2.6.2.1.2

Multiply $- a + x$ by $\frac{x - a}{| x - a |}$ .

$f'' (x) = - \frac{- | x - a | + \frac{(- a + x) (x - a)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.1.3

Simplify the numerator.

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Step 2.6.2.1.3.1

Reorder terms.

$f'' (x) = - \frac{- | x - a | + \frac{(- a + x) (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.1.3.2

Raise $- a + x$ to the power of $1$ .

$f'' (x) = - \frac{- | x - a | + \frac{(- a + x) (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.1.3.3

Raise $- a + x$ to the power of $1$ .

$f'' (x) = - \frac{- | x - a | + \frac{(- a + x) (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.1.3.4

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

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

Step 2.6.2.1.3.5

Add $1$ and $1$ .

$f'' (x) = - \frac{- | x - a | + \frac{{(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.2

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

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

Step 2.6.2.3

Combine $- | x - a |$ and $\frac{| x - a |}{| x - a |}$ .

$f'' (x) = - \frac{\frac{- | x - a | | x - a |}{| x - a |} + \frac{{(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{- | x - a | | x - a | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5

Simplify the numerator.

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Step 2.6.2.5.1

Multiply $- | x - a | | x - a |$ .

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Step 2.6.2.5.1.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = - \frac{\frac{- | (x - a) (x - a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.1.2

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

$f'' (x) = - \frac{\frac{- | (x - a) (x - a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.1.3

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

$f'' (x) = - \frac{\frac{- | (x - a) (x - a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.1.4

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

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

Step 2.6.2.5.1.5

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{- | {(x - a)}^{2} | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.2

Rewrite ${(x - a)}^{2}$ as $(x - a) (x - a)$ .

$f'' (x) = - \frac{\frac{- | (x - a) (x - a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.3

Expand $(x - a) (x - a)$ using the FOIL Method.

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Step 2.6.2.5.3.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{- | x (x - a) - a (x - a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.3.2

Apply the distributive property.

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

Step 2.6.2.5.3.3

Apply the distributive property.

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

Step 2.6.2.5.4

Simplify and combine like terms.

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Step 2.6.2.5.4.1

Simplify each term.

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Step 2.6.2.5.4.1.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{- | x^{2} + x (- a) - a x - a (- a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.4.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{- | x^{2} - x a - a x - a (- a) | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.4.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.6.2.5.4.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.6.2.5.4.1.4.1

Move $a$ .

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

Step 2.6.2.5.4.1.4.2

Multiply $a$ by $a$ .

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

Step 2.6.2.5.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.6.2.5.4.1.6

Multiply $a^{2}$ by $1$ .

$f'' (x) = - \frac{\frac{- | x^{2} - x a - a x + a^{2} | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.4.2

Subtract $a x$ from $- x a$ .

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Step 2.6.2.5.4.2.1

Move $x$ .

$f'' (x) = - \frac{\frac{- | x^{2} - a x - a x + a^{2} | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.4.2.2

Subtract $a x$ from $- a x$ .

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | + {(- a + x)}^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.5

Rewrite ${(- a + x)}^{2}$ as $(- a + x) (- a + x)$ .

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | + (- a + x) (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.6

Expand $(- a + x) (- a + x)$ using the FOIL Method.

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Step 2.6.2.5.6.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | - a (- a + x) + x (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.6.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | - a (- a) - a x + x (- a + x)}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.6.3

Apply the distributive property.

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

Step 2.6.2.5.7

Simplify and combine like terms.

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Step 2.6.2.5.7.1

Simplify each term.

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Step 2.6.2.5.7.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.6.2.5.7.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.6.2.5.7.1.2.1

Move $a$ .

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

Step 2.6.2.5.7.1.2.2

Multiply $a$ by $a$ .

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

Step 2.6.2.5.7.1.3

Multiply $- 1$ by $- 1$ .

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

Step 2.6.2.5.7.1.4

Multiply $a^{2}$ by $1$ .

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

Step 2.6.2.5.7.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.6.2.5.7.1.6

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | + a^{2} - a x - x a + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.7.2

Subtract $x a$ from $- a x$ .

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Step 2.6.2.5.7.2.1

Move $x$ .

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | + a^{2} - a x - a x + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.5.7.2.2

Subtract $a x$ from $- a x$ .

$f'' (x) = - \frac{\frac{- | x^{2} - 2 a x + a^{2} | + a^{2} - 2 a x + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.6

Factor $- 1$ out of $- | x^{2} - 2 a x + a^{2} |$ .

$f'' (x) = - \frac{\frac{- (| x^{2} - 2 a x + a^{2} |) + a^{2} - 2 a x + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.7

Factor $- 1$ out of $a^{2}$ .

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

Step 2.6.2.8

Factor $- 1$ out of $- (| x^{2} - 2 a x + a^{2} |) - 1 (- a^{2})$ .

$f'' (x) = - \frac{\frac{- (| x^{2} - 2 a x + a^{2} | - a^{2}) - 2 a x + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.9

Factor $- 1$ out of $- 2 a x$ .

$f'' (x) = - \frac{\frac{- (| x^{2} - 2 a x + a^{2} | - a^{2}) - (2 a x) + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.10

Factor $- 1$ out of $- (| x^{2} - 2 a x + a^{2} | - a^{2}) - (2 a x)$ .

$f'' (x) = - \frac{\frac{- (| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x) + x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.11

Factor $- 1$ out of $x^{2}$ .

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

Step 2.6.2.12

Factor $- 1$ out of $- (| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x) - 1 (- x^{2})$ .

$f'' (x) = - \frac{\frac{- (| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2})}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.2.13

Rewrite $- (| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2})$ as $- 1 (| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2})$ .

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

Step 2.6.2.14

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{| x - a |}}{{| x - a |}^{2}}$

Step 2.6.3

Combine terms.

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Step 2.6.3.1

Rewrite $\frac{- \frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{| x - a |}}{{| x - a |}^{2}}$ as a product.

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

Step 2.6.3.2

Multiply $\frac{1}{{| x - a |}^{2}}$ by $\frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{| x - a |}$ .

$f'' (x) = \frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{{| x - a |}^{2} | x - a |}$

Step 2.6.3.3

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

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Step 2.6.3.3.1

Multiply ${| x - a |}^{2}$ by $| x - a |$ .

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Step 2.6.3.3.1.1

Raise $| x - a |$ to the power of $1$ .

$f'' (x) = \frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{{| x - a |}^{2} | x - a |}$

Step 2.6.3.3.1.2

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

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

Step 2.6.3.3.2

Add $2$ and $1$ .

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

Step 2.6.3.4

Multiply $- 1$ by $- 1$ .

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

Step 2.6.3.5

Multiply $\frac{| x^{2} - 2 a x + a^{2} | - a^{2} + 2 a x - x^{2}}{{| x - a |}^{3}}$ by $1$ .

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

Step 2.6.4

Reorder terms.

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{a - x}{| x - a |} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $| x - a | - b$ with respect to $x$ is $\frac{d}{d x} [| x - a |] + \frac{d}{d x} [- b]$ .

$\frac{d}{d x} [| x - a |] + \frac{d}{d x} [- b]$

Step 4.1.2

Evaluate $\frac{d}{d x} [| x - a |]$ .

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Step 4.1.2.1

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 $g (x) = x - a$ .

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Step 4.1.2.1.1

To apply the Chain Rule, set $u$ as $x - a$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 4.1.2.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 4.1.2.1.3

Replace all occurrences of $u$ with $x - a$ .

$\frac{x - a}{| x - a |} \frac{d}{d x} [x - a] + \frac{d}{d x} [- b]$

Step 4.1.2.2

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

$\frac{x - a}{| x - a |} (\frac{d}{d x} [x] + \frac{d}{d x} [- a]) + \frac{d}{d x} [- b]$

Step 4.1.2.3

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

$\frac{x - a}{| x - a |} (1 + \frac{d}{d x} [- a]) + \frac{d}{d x} [- b]$

Step 4.1.2.4

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

$\frac{x - a}{| x - a |} (1 + 0) + \frac{d}{d x} [- b]$

Step 4.1.2.5

Add $1$ and $0$ .

$\frac{x - a}{| x - a |} \cdot 1 + \frac{d}{d x} [- b]$

Step 4.1.2.6

Multiply $\frac{x - a}{| x - a |}$ by $1$ .

$\frac{x - a}{| x - a |} + \frac{d}{d x} [- b]$

Step 4.1.3

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

$\frac{x - a}{| x - a |} + 0$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Add $\frac{x - a}{| x - a |}$ and $0$ .

$\frac{x - a}{| x - a |}$

Step 4.1.4.2

Reorder terms.

$\frac{- a + x}{| x - a |}$

Step 4.1.4.3

Factor $- 1$ out of $- a$ .

$\frac{- (a) + x}{| x - a |}$

Step 4.1.4.4

Factor $- 1$ out of $x$ .

$\frac{- (a) - 1 (- x)}{| x - a |}$

Step 4.1.4.5

Factor $- 1$ out of $- (a) - 1 (- x)$ .

$\frac{- (a - x)}{| x - a |}$

Step 4.1.4.6

Rewrite $- (a - x)$ as $- 1 (a - x)$ .

$\frac{- 1 (a - x)}{| x - a |}$

Step 4.1.4.7

Move the negative in front of the fraction.

$f' (x) = - \frac{a - x}{| x - a |}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{a - x}{| x - a |}$ .

$- \frac{a - x}{| x - a |}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{a - x}{| x - a |} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{a - x}{| x - a |} = 0$

Step 5.2

Set the numerator equal to zero.

$a - x = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Subtract $a$ from both sides of the equation.

$- x = - a$

Step 5.3.2

Divide each term in $- x = - a$ by $- 1$ and simplify.

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Step 5.3.2.1

Divide each term in $- x = - a$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- a}{- 1}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- a}{- 1}$

Step 5.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- a}{- 1}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{a}{1}$

Step 5.3.2.3.2

Divide $a$ by $1$ .

$x = a$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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 7

Critical points to evaluate.

$x = a$

Step 8

Evaluate the second derivative at $x = a$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{- a^{2} - {(a)}^{2} + 2 a (a) + | {(a)}^{2} - 2 a \cdot a + a^{2} |}{{| (a) - a |}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Subtract $a$ from $a$ .

$\frac{- a^{2} - {(a)}^{2} + 2 a (a) + | {(a)}^{2} - 2 a \cdot a + a^{2} |}{{| 0 |}^{3}}$

Step 9.2

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

$\frac{- a^{2} - {(a)}^{2} + 2 a (a) + | {(a)}^{2} - 2 a \cdot a + a^{2} |}{0^{3}}$

Step 9.3

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

$\frac{- a^{2} - {(a)}^{2} + 2 a (a) + | {(a)}^{2} - 2 a \cdot a + a^{2} |}{0}$

Step 9.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11