Calculus Examples

$f (x) = \frac{x^{2} - 2 x}{x^{2} + 2 x - 8}$

Step 1

Find the first derivative.

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Step 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 $f (x) = x^{2} - 2 x$ and $g (x) = x^{2} + 2 x - 8$ .

$\frac{(x^{2} + 2 x - 8) \frac{d}{d x} [x^{2} - 2 x] - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(x^{2} + 2 x - 8) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x]) - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.2

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

$\frac{(x^{2} + 2 x - 8) (2 x + \frac{d}{d x} [- 2 x]) - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$\frac{(x^{2} + 2 x - 8) (2 x - 2 \frac{d}{d x} [x]) - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.4

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

$\frac{(x^{2} + 2 x - 8) (2 x - 2 \cdot 1) - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.5

Multiply $- 2$ by $1$ .

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) \frac{d}{d x} [x^{2} + 2 x - 8]}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.6

By the Sum Rule, the derivative of $x^{2} + 2 x - 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 8]$ .

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 8])}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.7

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

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 8])}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.8

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.9

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

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (2 x + 2 \cdot 1 + \frac{d}{d x} [- 8])}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.10

Multiply $2$ by $1$ .

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (2 x + 2 + \frac{d}{d x} [- 8])}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.11

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

$\frac{(x^{2} + 2 x - 8) (2 x - 2) - (x^{2} - 2 x) (2 x + 2 + 0)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.2.12

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Expand $(x^{2} + 2 x - 8) (2 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.3.2.1.2

Simplify each term.

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Step 1.3.2.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2.1.2.2

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

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Step 1.3.2.1.2.2.1

Move $x$ .

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

Step 1.3.2.1.2.2.2

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

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Step 1.3.2.1.2.2.2.1

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

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

Step 1.3.2.1.2.2.2.2

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

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

Step 1.3.2.1.2.2.3

Add $1$ and $2$ .

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

Step 1.3.2.1.2.3

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

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

Step 1.3.2.1.2.4

Rewrite using the commutative property of multiplication.

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

Step 1.3.2.1.2.5

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

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Step 1.3.2.1.2.5.1

Move $x$ .

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

Step 1.3.2.1.2.5.2

Multiply $x$ by $x$ .

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

Step 1.3.2.1.2.6

Multiply $2$ by $2$ .

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

Step 1.3.2.1.2.7

Multiply $- 2$ by $2$ .

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

Step 1.3.2.1.2.8

Multiply $2$ by $- 8$ .

$\frac{2 x^{3} - 2 x^{2} + 4 x^{2} - 4 x - 16 x - 8 \cdot - 2 + (- x^{2} - (- 2 x)) (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.2.9

Multiply $- 8$ by $- 2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 x^{2} - 4 x - 16 x + 16 + (- x^{2} - (- 2 x)) (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.3

Add $- 2 x^{2}$ and $4 x^{2}$ .

$\frac{2 x^{3} + 2 x^{2} - 4 x - 16 x + 16 + (- x^{2} - (- 2 x)) (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.4

Subtract $16 x$ from $- 4 x$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 + (- x^{2} - (- 2 x)) (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.5

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 + (- x^{2} + 2 x) (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.6

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

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Step 1.3.2.1.6.1

Apply the distributive property.

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - x^{2} (2 x + 2) + 2 x (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.6.2

Apply the distributive property.

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - x^{2} (2 x) - x^{2} \cdot 2 + 2 x (2 x + 2)}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.6.3

Apply the distributive property.

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - x^{2} (2 x) - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7

Simplify and combine like terms.

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Step 1.3.2.1.7.1

Simplify each term.

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Step 1.3.2.1.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 1 \cdot 2 x^{2} x - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.2

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

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Step 1.3.2.1.7.1.2.1

Move $x$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.2.2

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

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Step 1.3.2.1.7.1.2.2.1

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

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.2.2.2

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

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 1 \cdot 2 x^{3} - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - x^{2} \cdot 2 + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.4

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 2 x (2 x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 2 \cdot 2 x \cdot x + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.6

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

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Step 1.3.2.1.7.1.6.1

Move $x$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 2 \cdot 2 (x \cdot x) + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 2 \cdot 2 x^{2} + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.7

Multiply $2$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 4 x^{2} + 2 x \cdot 2}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.1.8

Multiply $2$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} - 2 x^{2} + 4 x^{2} + 4 x}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.1.7.2

Add $- 2 x^{2}$ and $4 x^{2}$ .

$\frac{2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} + 2 x^{2} + 4 x}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.2

Combine the opposite terms in $2 x^{3} + 2 x^{2} - 20 x + 16 - 2 x^{3} + 2 x^{2} + 4 x$ .

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Step 1.3.2.2.1

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

$\frac{2 x^{2} - 20 x + 16 + 0 + 2 x^{2} + 4 x}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.2.2

Add $2 x^{2} - 20 x + 16$ and $0$ .

$\frac{2 x^{2} - 20 x + 16 + 2 x^{2} + 4 x}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.3

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

$\frac{4 x^{2} - 20 x + 16 + 4 x}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.2.4

Add $- 20 x$ and $4 x$ .

$\frac{4 x^{2} - 16 x + 16}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Factor $4$ out of $4 x^{2} - 16 x + 16$ .

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Step 1.3.3.1.1

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

$\frac{4 (x^{2}) - 16 x + 16}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.3.1.2

Factor $4$ out of $- 16 x$ .

$\frac{4 (x^{2}) + 4 (- 4 x) + 16}{{(x^{2} + 2 x - 8)}^{2}}$

Step 1.3.3.1.3

Factor $4$ out of $16$ .

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

Step 1.3.3.1.4

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

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

Step 1.3.3.1.5

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

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

Step 1.3.3.2

Factor using the perfect square rule.

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Step 1.3.3.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 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.

$4 x = 2 \cdot x \cdot 2$

Step 1.3.3.2.3

Rewrite the polynomial.

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

Step 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 $b = 2$ .

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

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.3.4.1.2

Write the factored form using these integers.

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

Step 1.3.4.2

Apply the product rule to $(x - 2) (x + 4)$ .

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

Step 1.3.5

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

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Step 1.3.5.1

Cancel the common factor.

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

Step 1.3.5.2

Rewrite the expression.

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 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 2.1.2

Apply basic rules of exponents.

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Step 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 2.1.2.2

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

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Step 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 2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 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 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 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 2.2.3

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

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $4$ .

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

Step 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 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 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 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

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

Step 2.3.5.2

Multiply $- 8$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Move the negative in front of the fraction.

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

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8}{{(x + 4)}^{3}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{1}{{(x + 4)}^{3}}]$ .

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

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(x + 4)}^{3}}$ as ${({(x + 4)}^{3})}^{- 1}$ .

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

Step 3.1.2.2

Multiply the exponents in ${({(x + 4)}^{3})}^{- 1}$ .

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = x + 4$ .

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Step 3.2.1

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

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 3$ .

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

Step 3.2.3

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

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 3$ by $- 8$ .

$24 ({(x + 4)}^{- 4} \frac{d}{d x} [x + 4])$

Step 3.3.2

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

$24 {(x + 4)}^{- 4} (\frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 3.3.3

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

$24 {(x + 4)}^{- 4} (1 + \frac{d}{d x} [4])$

Step 3.3.4

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

$24 {(x + 4)}^{- 4} (1 + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$24 {(x + 4)}^{- 4} \cdot 1$

Step 3.3.5.2

Multiply $24$ by $1$ .

$24 {(x + 4)}^{- 4}$

Step 3.4

Simplify.

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Step 3.4.1

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

$24 \frac{1}{{(x + 4)}^{4}}$

Step 3.4.2

Combine $24$ and $\frac{1}{{(x + 4)}^{4}}$ .

$f''' (x) = \frac{24}{{(x + 4)}^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Since $24$ is constant with respect to $x$ , the derivative of $\frac{24}{{(x + 4)}^{4}}$ with respect to $x$ is $24 \frac{d}{d x} [\frac{1}{{(x + 4)}^{4}}]$ .

$24 \frac{d}{d x} [\frac{1}{{(x + 4)}^{4}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

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

$24 \frac{d}{d x} [{({(x + 4)}^{4})}^{- 1}]$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$24 \frac{d}{d x} [{(x + 4)}^{4 \cdot - 1}]$

Step 4.1.2.2.2

Multiply $4$ by $- 1$ .

$24 \frac{d}{d x} [{(x + 4)}^{- 4}]$

Step 4.2

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

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Step 4.2.1

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

$24 (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [x + 4])$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 4$ .

$24 (- 4 u^{- 5} \frac{d}{d x} [x + 4])$

Step 4.2.3

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

$24 (- 4 {(x + 4)}^{- 5} \frac{d}{d x} [x + 4])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 4$ by $24$ .

$- 96 ({(x + 4)}^{- 5} \frac{d}{d x} [x + 4])$

Step 4.3.2

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

$- 96 {(x + 4)}^{- 5} (\frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 4.3.3

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

$- 96 {(x + 4)}^{- 5} (1 + \frac{d}{d x} [4])$

Step 4.3.4

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

$- 96 {(x + 4)}^{- 5} (1 + 0)$

Step 4.3.5

Simplify the expression.

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Step 4.3.5.1

Add $1$ and $0$ .

$- 96 {(x + 4)}^{- 5} \cdot 1$

Step 4.3.5.2

Multiply $- 96$ by $1$ .

$- 96 {(x + 4)}^{- 5}$

Step 4.4

Simplify.

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Step 4.4.1

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

$- 96 \frac{1}{{(x + 4)}^{5}}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Combine $- 96$ and $\frac{1}{{(x + 4)}^{5}}$ .

$\frac{- 96}{{(x + 4)}^{5}}$

Step 4.4.2.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{96}{{(x + 4)}^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{96}{{(x + 4)}^{5}}$ .

$- \frac{96}{{(x + 4)}^{5}}$