Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = x^{2} - x$ and $g (x) = x^{2} + 3 x - 4$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

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

$\frac{(x^{2} + 3 x - 4) (2 x - 1 \cdot 1) - (x^{2} - x) \frac{d}{d x} [x^{2} + 3 x - 4]}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.5

Multiply $- 1$ by $1$ .

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) \frac{d}{d x} [x^{2} + 3 x - 4]}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.6

By the Sum Rule, the derivative of $x^{2} + 3 x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 4])}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.7

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

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 4])}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.8

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

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.9

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

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + 3 \cdot 1 + \frac{d}{d x} [- 4])}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.10

Multiply $3$ by $1$ .

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + 3 + \frac{d}{d x} [- 4])}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.11

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

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + 3 + 0)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.2.12

Add $2 x + 3$ and $0$ .

$\frac{(x^{2} + 3 x - 4) (2 x - 1) - (x^{2} - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

$\frac{(x^{2} + 3 x - 4) (2 x - 1) + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2

Simplify the numerator.

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Step 1.1.1.3.2.1

Simplify each term.

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Step 1.1.1.3.2.1.1

Expand $(x^{2} + 3 x - 4) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2

Simplify each term.

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Step 1.1.1.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.2

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

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Step 1.1.1.3.2.1.2.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.2.2

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

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Step 1.1.1.3.2.1.2.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.3

Move $- 1$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 1 \cdot x^{2} + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{2 x^{3} - x^{2} + 3 x (2 x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - x^{2} + 3 \cdot 2 x \cdot x + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.3.2.1.2.6.1

Move $x$ .

$\frac{2 x^{3} - x^{2} + 3 \cdot 2 (x \cdot x) + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - x^{2} + 3 \cdot 2 x^{2} + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.7

Multiply $3$ by $2$ .

$\frac{2 x^{3} - x^{2} + 6 x^{2} + 3 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.8

Multiply $- 1$ by $3$ .

$\frac{2 x^{3} - x^{2} + 6 x^{2} - 3 x - 4 (2 x) - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.9

Multiply $2$ by $- 4$ .

$\frac{2 x^{3} - x^{2} + 6 x^{2} - 3 x - 8 x - 4 \cdot - 1 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.2.10

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{3} - x^{2} + 6 x^{2} - 3 x - 8 x + 4 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.3

Add $- x^{2}$ and $6 x^{2}$ .

$\frac{2 x^{3} + 5 x^{2} - 3 x - 8 x + 4 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.4

Subtract $8 x$ from $- 3 x$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 + (- x^{2} - - x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.5

Multiply $- - x$ .

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Step 1.1.1.3.2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 + (- x^{2} + 1 x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.5.2

Multiply $x$ by $1$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 + (- x^{2} + x) (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.6

Expand $(- x^{2} + x) (2 x + 3)$ using the FOIL Method.

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Step 1.1.1.3.2.1.6.1

Apply the distributive property.

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - x^{2} (2 x + 3) + x (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.6.2

Apply the distributive property.

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - x^{2} (2 x) - x^{2} \cdot 3 + x (2 x + 3)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.6.3

Apply the distributive property.

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - x^{2} (2 x) - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7

Simplify and combine like terms.

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Step 1.1.1.3.2.1.7.1

Simplify each term.

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Step 1.1.1.3.2.1.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 1 \cdot 2 x^{2} x - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.2

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

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Step 1.1.1.3.2.1.7.1.2.1

Move $x$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.2.2

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

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Step 1.1.1.3.2.1.7.1.2.2.1

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

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.2.2.2

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

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 1 \cdot 2 x^{3} - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - x^{2} \cdot 3 + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.4

Multiply $3$ by $- 1$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - 3 x^{2} + x (2 x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - 3 x^{2} + 2 x \cdot x + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.3.2.1.7.1.6.1

Move $x$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - 3 x^{2} + 2 (x \cdot x) + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - 3 x^{2} + 2 x^{2} + x \cdot 3}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.1.7

Move $3$ to the left of $x$ .

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - 3 x^{2} + 2 x^{2} + 3 x}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.1.7.2

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

$\frac{2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - x^{2} + 3 x}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.2

Combine the opposite terms in $2 x^{3} + 5 x^{2} - 11 x + 4 - 2 x^{3} - x^{2} + 3 x$ .

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Step 1.1.1.3.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{5 x^{2} - 11 x + 4 + 0 - x^{2} + 3 x}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.2.2

Add $5 x^{2} - 11 x + 4$ and $0$ .

$\frac{5 x^{2} - 11 x + 4 - x^{2} + 3 x}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.3

Subtract $x^{2}$ from $5 x^{2}$ .

$\frac{4 x^{2} - 11 x + 4 + 3 x}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.2.4

Add $- 11 x$ and $3 x$ .

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

Step 1.1.1.3.3

Simplify the numerator.

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Step 1.1.1.3.3.1

Factor $4$ out of $4 x^{2} - 8 x + 4$ .

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Step 1.1.1.3.3.1.1

Factor $4$ out of $4 x^{2}$ .

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

Step 1.1.1.3.3.1.2

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

$\frac{4 (x^{2}) + 4 (- 2 x) + 4}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.1.3

Factor $4$ out of $4$ .

$\frac{4 (x^{2}) + 4 (- 2 x) + 4 (1)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.1.4

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

$\frac{4 (x^{2} - 2 x) + 4 (1)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.1.5

Factor $4$ out of $4 (x^{2} - 2 x) + 4 (1)$ .

$\frac{4 (x^{2} - 2 x + 1)}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.2

Factor using the perfect square rule.

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Step 1.1.1.3.3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{4 (x^{2} - 2 x + 1^{2})}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.1.1.3.3.2.3

Rewrite the polynomial.

$\frac{4 (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{4 {(x - 1)}^{2}}{{(x^{2} + 3 x - 4)}^{2}}$

Step 1.1.1.3.4

Simplify the denominator.

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Step 1.1.1.3.4.1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 1.1.1.3.4.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.1.1.3.4.1.2

Write the factored form using these integers.

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

Step 1.1.1.3.4.2

Apply the product rule to $(x - 1) (x + 4)$ .

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

Step 1.1.1.3.5

Cancel the common factor of ${(x - 1)}^{2}$ .

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Step 1.1.1.3.5.1

Cancel the common factor.

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

Step 1.1.1.3.5.2

Rewrite the expression.

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.2.1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{{(x + 4)}^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{{(x + 4)}^{2}}]$ .

$4 \frac{d}{d x} [\frac{1}{{(x + 4)}^{2}}]$

Step 1.1.2.1.2

Apply basic rules of exponents.

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Step 1.1.2.1.2.1

Rewrite $\frac{1}{{(x + 4)}^{2}}$ as ${({(x + 4)}^{2})}^{- 1}$ .

$4 \frac{d}{d x} [{({(x + 4)}^{2})}^{- 1}]$

Step 1.1.2.1.2.2

Multiply the exponents in ${({(x + 4)}^{2})}^{- 1}$ .

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Step 1.1.2.1.2.2.1

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

$4 \frac{d}{d x} [{(x + 4)}^{2 \cdot - 1}]$

Step 1.1.2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = x + 4$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $x + 4$ .

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

Replace all occurrences of $u$ with $x + 4$ .

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

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

Multiply $- 2$ by $4$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

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

$- 8 {(x + 4)}^{- 3} (1 + \frac{d}{d x} [4])$

Step 1.1.2.3.4

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

$- 8 {(x + 4)}^{- 3} (1 + 0)$

Step 1.1.2.3.5

Simplify the expression.

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Step 1.1.2.3.5.1

Add $1$ and $0$ .

$- 8 {(x + 4)}^{- 3} \cdot 1$

Step 1.1.2.3.5.2

Multiply $- 8$ by $1$ .

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

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8 \frac{1}{{(x + 4)}^{3}}$

Step 1.1.2.4.2

Combine terms.

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Step 1.1.2.4.2.1

Combine $- 8$ and $\frac{1}{{(x + 4)}^{3}}$ .

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

Step 1.1.2.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.3

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

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

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{8}{{(x + 4)}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{8}{{(x + 4)}^{3}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$8 = 0$

Step 1.2.3

Since $8 \neq 0$ , there are no solutions.

No solution

Step 2

Find the domain of $f (x) = \frac{x^{2} - x}{x^{2} + 3 x - 4}$ .

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Step 2.1

Set the denominator in $\frac{x^{2} - x}{x^{2} + 3 x - 4}$ equal to $0$ to find where the expression is undefined.

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

Step 2.2

Solve for $x$ .

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Step 2.2.1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 2.2.1.2

Write the factored form using these integers.

$(x - 1) (x + 4) = 0$

Step 2.2.2

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

$x - 1 = 0$

$x + 4 = 0$

Step 2.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.2.5

The final solution is all the values that make $(x - 1) (x + 4) = 0$ true.

$x = 1, - 4$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 4}$

Interval Notation:

$(- \infty, - 4) \cup (- 4, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 4}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 4) \cup (- 4, 1) \cup (1, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 6$ in the expression.

$f'' (- 6) = - \frac{8}{{((- 6) + 4)}^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the denominator.

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Step 4.2.1.1

Add $- 6$ and $4$ .

$f'' (- 6) = - \frac{8}{{(- 2)}^{3}}$

Step 4.2.1.2

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

$f'' (- 6) = - \frac{8}{- 8}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Divide $8$ by $- 8$ .

$f'' (- 6) = 1$

Step 4.2.2.2

Multiply $- 1$ by $- 1$ .

$f'' (- 6) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 4.3

The graph is concave up on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 4, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - \frac{8}{{((0) + 4)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the denominator.

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Step 5.2.1.1

Add $0$ and $4$ .

$f'' (0) = - \frac{8}{4^{3}}$

Step 5.2.1.2

Raise $4$ to the power of $3$ .

$f'' (0) = - \frac{8}{64}$

Step 5.2.2

Cancel the common factor of $8$ and $64$ .

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Step 5.2.2.1

Factor $8$ out of $8$ .

$f'' (0) = - \frac{8 (1)}{64}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $8$ out of $64$ .

$f'' (0) = - \frac{8 \cdot 1}{8 \cdot 8}$

Step 5.2.2.2.2

Cancel the common factor.

$f'' (0) = - \frac{8 \cdot 1}{8 \cdot 8}$

Step 5.2.2.2.3

Rewrite the expression.

$f'' (0) = - \frac{1}{8}$

Step 5.2.3

The final answer is $- \frac{1}{8}$ .

$- \frac{1}{8}$

Step 5.3

The graph is concave down on the interval $(- 4, 1)$ because $f'' (0)$ is negative.

Concave down on $(- 4, 1)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Add $4$ and $4$ .

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

Step 6.2.1.2

Raise $8$ to the power of $3$ .

$f'' (4) = - \frac{8}{512}$

Step 6.2.2

Cancel the common factor of $8$ and $512$ .

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Step 6.2.2.1

Factor $8$ out of $8$ .

$f'' (4) = - \frac{8 (1)}{512}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $8$ out of $512$ .

$f'' (4) = - \frac{8 \cdot 1}{8 \cdot 64}$

Step 6.2.2.2.2

Cancel the common factor.

$f'' (4) = - \frac{8 \cdot 1}{8 \cdot 64}$

Step 6.2.2.2.3

Rewrite the expression.

$f'' (4) = - \frac{1}{64}$

Step 6.2.3

The final answer is $- \frac{1}{64}$ .

$- \frac{1}{64}$

Step 6.3

The graph is concave down on the interval $(1, \infty)$ because $f'' (4)$ is negative.

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Concave down on $(- 4, 1)$ since $f'' (x)$ is negative

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 8