Calculus Examples

$f (x) = (x^{3} + 4 x^{2} - x + 5) (x^{5} - 6 x^{4} - 7)$

Step 1

Find the first derivative.

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Step 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 $f (x) = x^{3} + 4 x^{2} - x + 5$ and $g (x) = x^{5} - 6 x^{4} - 7$ .

$(x^{3} + 4 x^{2} - x + 5) \frac{d}{d x} [x^{5} - 6 x^{4} - 7] + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{5} - 6 x^{4} - 7$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 6 x^{4}] + \frac{d}{d x} [- 7]$ .

$(x^{3} + 4 x^{2} - x + 5) (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 6 x^{4}] + \frac{d}{d x} [- 7]) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} + \frac{d}{d x} [- 6 x^{4}] + \frac{d}{d x} [- 7]) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{4}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{4}]$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 6 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 7]) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.4

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 6 (4 x^{3}) + \frac{d}{d x} [- 7]) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.5

Multiply $4$ by $- 6$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3} + \frac{d}{d x} [- 7]) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3} + 0) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.7

Add $5 x^{4} - 24 x^{3}$ and $0$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) \frac{d}{d x} [x^{3} + 4 x^{2} - x + 5]$

Step 1.2.8

By the Sum Rule, the derivative of $x^{3} + 4 x^{2} - x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5])$

Step 1.2.9

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5])$

Step 1.2.10

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5])$

Step 1.2.11

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 4 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [5])$

Step 1.2.12

Multiply $2$ by $4$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x + \frac{d}{d x} [- x] + \frac{d}{d x} [5])$

Step 1.2.13

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 1.2.14

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

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1 \cdot 1 + \frac{d}{d x} [5])$

Step 1.2.15

Multiply $- 1$ by $1$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1 + \frac{d}{d x} [5])$

Step 1.2.16

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1 + 0)$

Step 1.2.17

Add $3 x^{2} + 8 x - 1$ and $0$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$

Step 1.3

Simplify.

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Step 1.3.1

Factor $1$ out of $(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$ .

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Step 1.3.1.1

Factor $1$ out of $(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3})$ .

$1 ((x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3})) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$

Step 1.3.1.2

Factor $1$ out of $(x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$ .

$1 ((x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3})) + 1 ((x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1))$

Step 1.3.1.3

Factor $1$ out of $1 ((x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3})) + 1 ((x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1))$ .

$1 ((x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1))$

Step 1.3.2

Multiply $(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$ by $1$ .

$(x^{3} + 4 x^{2} - x + 5) (5 x^{4} - 24 x^{3}) + (x^{5} - 6 x^{4} - 7) (3 x^{2} + 8 x - 1)$

Step 1.3.3

Reorder terms.

$(5 x^{4} - 24 x^{3}) (x^{3} + 4 x^{2} - x + 5) + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4

Simplify each term.

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Step 1.3.4.1

Expand $(5 x^{4} - 24 x^{3}) (x^{3} + 4 x^{2} - x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$5 x^{4} x^{3} + 5 x^{4} (4 x^{2}) + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2

Simplify each term.

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Step 1.3.4.2.1

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.3.4.2.1.1

Move $x^{3}$ .

$5 (x^{3} x^{4}) + 5 x^{4} (4 x^{2}) + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.1.2

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

$5 x^{3 + 4} + 5 x^{4} (4 x^{2}) + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.1.3

Add $3$ and $4$ .

$5 x^{7} + 5 x^{4} (4 x^{2}) + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.2

Rewrite using the commutative property of multiplication.

$5 x^{7} + 5 \cdot 4 x^{4} x^{2} + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.3

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.3.4.2.3.1

Move $x^{2}$ .

$5 x^{7} + 5 \cdot 4 (x^{2} x^{4}) + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.3.2

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

$5 x^{7} + 5 \cdot 4 x^{2 + 4} + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.3.3

Add $2$ and $4$ .

$5 x^{7} + 5 \cdot 4 x^{6} + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.4

Multiply $5$ by $4$ .

$5 x^{7} + 20 x^{6} + 5 x^{4} (- x) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.5

Rewrite using the commutative property of multiplication.

$5 x^{7} + 20 x^{6} + 5 \cdot - 1 x^{4} x + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.6

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.3.4.2.6.1

Move $x$ .

$5 x^{7} + 20 x^{6} + 5 \cdot - 1 (x \cdot x^{4}) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.6.2

Multiply $x$ by $x^{4}$ .

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Step 1.3.4.2.6.2.1

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

$5 x^{7} + 20 x^{6} + 5 \cdot - 1 (x^{1} x^{4}) + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.6.2.2

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

$5 x^{7} + 20 x^{6} + 5 \cdot - 1 x^{1 + 4} + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.6.3

Add $1$ and $4$ .

$5 x^{7} + 20 x^{6} + 5 \cdot - 1 x^{5} + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.7

Multiply $5$ by $- 1$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 5 x^{4} \cdot 5 - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.8

Multiply $5$ by $5$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{3} x^{3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.9

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 1.3.4.2.9.1

Move $x^{3}$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 (x^{3} x^{3}) - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.9.2

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

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{3 + 3} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.9.3

Add $3$ and $3$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 24 x^{3} (4 x^{2}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.10

Rewrite using the commutative property of multiplication.

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 24 \cdot 4 x^{3} x^{2} - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.11

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 1.3.4.2.11.1

Move $x^{2}$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 24 \cdot 4 (x^{2} x^{3}) - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.11.2

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

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 24 \cdot 4 x^{2 + 3} - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.11.3

Add $2$ and $3$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 24 \cdot 4 x^{5} - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.12

Multiply $- 24$ by $4$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 x^{3} (- x) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.13

Rewrite using the commutative property of multiplication.

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 \cdot - 1 x^{3} x - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.14

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.3.4.2.14.1

Move $x$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 \cdot - 1 (x \cdot x^{3}) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.14.2

Multiply $x$ by $x^{3}$ .

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Step 1.3.4.2.14.2.1

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

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 \cdot - 1 (x^{1} x^{3}) - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.14.2.2

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

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 \cdot - 1 x^{1 + 3} - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.14.3

Add $1$ and $3$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} - 24 \cdot - 1 x^{4} - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.15

Multiply $- 24$ by $- 1$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} + 24 x^{4} - 24 x^{3} \cdot 5 + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.2.16

Multiply $5$ by $- 24$ .

$5 x^{7} + 20 x^{6} - 5 x^{5} + 25 x^{4} - 24 x^{6} - 96 x^{5} + 24 x^{4} - 120 x^{3} + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.3

Subtract $24 x^{6}$ from $20 x^{6}$ .

$5 x^{7} - 4 x^{6} - 5 x^{5} + 25 x^{4} - 96 x^{5} + 24 x^{4} - 120 x^{3} + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.4

Subtract $96 x^{5}$ from $- 5 x^{5}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 25 x^{4} + 24 x^{4} - 120 x^{3} + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.5

Add $25 x^{4}$ and $24 x^{4}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + (3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$

Step 1.3.4.6

Expand $(3 x^{2} + 8 x - 1) (x^{5} - 6 x^{4} - 7)$ by multiplying each term in the first expression by each term in the second expression.

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{2} x^{5} + 3 x^{2} (- 6 x^{4}) + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7

Simplify each term.

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Step 1.3.4.7.1

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 1.3.4.7.1.1

Move $x^{5}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 (x^{5} x^{2}) + 3 x^{2} (- 6 x^{4}) + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.1.2

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{5 + 2} + 3 x^{2} (- 6 x^{4}) + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.1.3

Add $5$ and $2$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} + 3 x^{2} (- 6 x^{4}) + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.2

Rewrite using the commutative property of multiplication.

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} + 3 \cdot - 6 x^{2} x^{4} + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.3

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 1.3.4.7.3.1

Move $x^{4}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} + 3 \cdot - 6 (x^{4} x^{2}) + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.3.2

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} + 3 \cdot - 6 x^{4 + 2} + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.3.3

Add $4$ and $2$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} + 3 \cdot - 6 x^{6} + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.4

Multiply $3$ by $- 6$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} + 3 x^{2} \cdot - 7 + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.5

Multiply $- 7$ by $3$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x \cdot x^{5} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.6

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 1.3.4.7.6.1

Move $x^{5}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 (x^{5} x) + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.6.2

Multiply $x^{5}$ by $x$ .

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Step 1.3.4.7.6.2.1

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 (x^{5} x^{1}) + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.6.2.2

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{5 + 1} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.6.3

Add $5$ and $1$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 x (- 6 x^{4}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.7

Rewrite using the commutative property of multiplication.

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 \cdot - 6 x \cdot x^{4} + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.8

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 1.3.4.7.8.1

Move $x^{4}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 \cdot - 6 (x^{4} x) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.8.2

Multiply $x^{4}$ by $x$ .

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Step 1.3.4.7.8.2.1

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 \cdot - 6 (x^{4} x^{1}) + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.8.2.2

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

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 \cdot - 6 x^{4 + 1} + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.8.3

Add $4$ and $1$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} + 8 \cdot - 6 x^{5} + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.9

Multiply $8$ by $- 6$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} - 48 x^{5} + 8 x \cdot - 7 - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.10

Multiply $- 7$ by $8$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} - 48 x^{5} - 56 x - 1 x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.11

Rewrite $- 1 x^{5}$ as $- x^{5}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} - 48 x^{5} - 56 x - x^{5} - 1 (- 6 x^{4}) - 1 \cdot - 7$

Step 1.3.4.7.12

Multiply $- 6$ by $- 1$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} - 48 x^{5} - 56 x - x^{5} + 6 x^{4} - 1 \cdot - 7$

Step 1.3.4.7.13

Multiply $- 1$ by $- 7$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 18 x^{6} - 21 x^{2} + 8 x^{6} - 48 x^{5} - 56 x - x^{5} + 6 x^{4} + 7$

Step 1.3.4.8

Add $- 18 x^{6}$ and $8 x^{6}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 10 x^{6} - 21 x^{2} - 48 x^{5} - 56 x - x^{5} + 6 x^{4} + 7$

Step 1.3.4.9

Subtract $x^{5}$ from $- 48 x^{5}$ .

$5 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} + 3 x^{7} - 10 x^{6} - 21 x^{2} - 49 x^{5} - 56 x + 6 x^{4} + 7$

Step 1.3.5

Add $5 x^{7}$ and $3 x^{7}$ .

$8 x^{7} - 4 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} - 10 x^{6} - 21 x^{2} - 49 x^{5} - 56 x + 6 x^{4} + 7$

Step 1.3.6

Subtract $10 x^{6}$ from $- 4 x^{6}$ .

$8 x^{7} - 14 x^{6} - 101 x^{5} + 49 x^{4} - 120 x^{3} - 21 x^{2} - 49 x^{5} - 56 x + 6 x^{4} + 7$

Step 1.3.7

Subtract $49 x^{5}$ from $- 101 x^{5}$ .

$8 x^{7} - 14 x^{6} - 150 x^{5} + 49 x^{4} - 120 x^{3} - 21 x^{2} - 56 x + 6 x^{4} + 7$

Step 1.3.8

Add $49 x^{4}$ and $6 x^{4}$ .

$f' (x) = 8 x^{7} - 14 x^{6} - 150 x^{5} + 55 x^{4} - 120 x^{3} - 21 x^{2} - 56 x + 7$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $8 x^{7} - 14 x^{6} - 150 x^{5} + 55 x^{4} - 120 x^{3} - 21 x^{2} - 56 x + 7$ with respect to $x$ is $\frac{d}{d x} [8 x^{7}] + \frac{d}{d x} [- 14 x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [8 x^{7}] + \frac{d}{d x} [- 14 x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.2

Evaluate $\frac{d}{d x} [8 x^{7}]$ .

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Step 2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{7}$ with respect to $x$ is $8 \frac{d}{d x} [x^{7}]$ .

$8 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- 14 x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$8 (7 x^{6}) + \frac{d}{d x} [- 14 x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.2.3

Multiply $7$ by $8$ .

$56 x^{6} + \frac{d}{d x} [- 14 x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.3

Evaluate $\frac{d}{d x} [- 14 x^{6}]$ .

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Step 2.3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x^{6}$ with respect to $x$ is $- 14 \frac{d}{d x} [x^{6}]$ .

$56 x^{6} - 14 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$56 x^{6} - 14 (6 x^{5}) + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.3.3

Multiply $6$ by $- 14$ .

$56 x^{6} - 84 x^{5} + \frac{d}{d x} [- 150 x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.4

Evaluate $\frac{d}{d x} [- 150 x^{5}]$ .

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Step 2.4.1

Since $- 150$ is constant with respect to $x$ , the derivative of $- 150 x^{5}$ with respect to $x$ is $- 150 \frac{d}{d x} [x^{5}]$ .

$56 x^{6} - 84 x^{5} - 150 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$56 x^{6} - 84 x^{5} - 150 (5 x^{4}) + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.4.3

Multiply $5$ by $- 150$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + \frac{d}{d x} [55 x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.5

Evaluate $\frac{d}{d x} [55 x^{4}]$ .

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Step 2.5.1

Since $55$ is constant with respect to $x$ , the derivative of $55 x^{4}$ with respect to $x$ is $55 \frac{d}{d x} [x^{4}]$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 55 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.5.2

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

$56 x^{6} - 84 x^{5} - 750 x^{4} + 55 (4 x^{3}) + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.5.3

Multiply $4$ by $55$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} + \frac{d}{d x} [- 120 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.6

Evaluate $\frac{d}{d x} [- 120 x^{3}]$ .

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Step 2.6.1

Since $- 120$ is constant with respect to $x$ , the derivative of $- 120 x^{3}$ with respect to $x$ is $- 120 \frac{d}{d x} [x^{3}]$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 120 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.6.2

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

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 120 (3 x^{2}) + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.6.3

Multiply $3$ by $- 120$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.7

Evaluate $\frac{d}{d x} [- 21 x^{2}]$ .

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Step 2.7.1

Since $- 21$ is constant with respect to $x$ , the derivative of $- 21 x^{2}$ with respect to $x$ is $- 21 \frac{d}{d x} [x^{2}]$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 21 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.7.2

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

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 21 (2 x) + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.7.3

Multiply $2$ by $- 21$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x + \frac{d}{d x} [- 56 x] + \frac{d}{d x} [7]$

Step 2.8

Evaluate $\frac{d}{d x} [- 56 x]$ .

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Step 2.8.1

Since $- 56$ is constant with respect to $x$ , the derivative of $- 56 x$ with respect to $x$ is $- 56 \frac{d}{d x} [x]$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 2.8.2

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

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56 \cdot 1 + \frac{d}{d x} [7]$

Step 2.8.3

Multiply $- 56$ by $1$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56 + \frac{d}{d x} [7]$

Step 2.9

Differentiate using the Constant Rule.

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Step 2.9.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56 + 0$

Step 2.9.2

Add $56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56$ and $0$ .

$f'' (x) = 56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $56 x^{6} - 84 x^{5} - 750 x^{4} + 220 x^{3} - 360 x^{2} - 42 x - 56$ with respect to $x$ is $\frac{d}{d x} [56 x^{6}] + \frac{d}{d x} [- 84 x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$ .

$\frac{d}{d x} [56 x^{6}] + \frac{d}{d x} [- 84 x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.2

Evaluate $\frac{d}{d x} [56 x^{6}]$ .

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Step 3.2.1

Since $56$ is constant with respect to $x$ , the derivative of $56 x^{6}$ with respect to $x$ is $56 \frac{d}{d x} [x^{6}]$ .

$56 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 84 x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$56 (6 x^{5}) + \frac{d}{d x} [- 84 x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.2.3

Multiply $6$ by $56$ .

$336 x^{5} + \frac{d}{d x} [- 84 x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.3

Evaluate $\frac{d}{d x} [- 84 x^{5}]$ .

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Step 3.3.1

Since $- 84$ is constant with respect to $x$ , the derivative of $- 84 x^{5}$ with respect to $x$ is $- 84 \frac{d}{d x} [x^{5}]$ .

$336 x^{5} - 84 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$336 x^{5} - 84 (5 x^{4}) + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.3.3

Multiply $5$ by $- 84$ .

$336 x^{5} - 420 x^{4} + \frac{d}{d x} [- 750 x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.4

Evaluate $\frac{d}{d x} [- 750 x^{4}]$ .

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Step 3.4.1

Since $- 750$ is constant with respect to $x$ , the derivative of $- 750 x^{4}$ with respect to $x$ is $- 750 \frac{d}{d x} [x^{4}]$ .

$336 x^{5} - 420 x^{4} - 750 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.4.2

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

$336 x^{5} - 420 x^{4} - 750 (4 x^{3}) + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.4.3

Multiply $4$ by $- 750$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + \frac{d}{d x} [220 x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.5

Evaluate $\frac{d}{d x} [220 x^{3}]$ .

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Step 3.5.1

Since $220$ is constant with respect to $x$ , the derivative of $220 x^{3}$ with respect to $x$ is $220 \frac{d}{d x} [x^{3}]$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 220 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.5.2

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

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 220 (3 x^{2}) + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.5.3

Multiply $3$ by $220$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} + \frac{d}{d x} [- 360 x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.6

Evaluate $\frac{d}{d x} [- 360 x^{2}]$ .

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Step 3.6.1

Since $- 360$ is constant with respect to $x$ , the derivative of $- 360 x^{2}$ with respect to $x$ is $- 360 \frac{d}{d x} [x^{2}]$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 360 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.6.2

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

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 360 (2 x) + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.6.3

Multiply $2$ by $- 360$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [- 56]$

Step 3.7

Evaluate $\frac{d}{d x} [- 42 x]$ .

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Step 3.7.1

Since $- 42$ is constant with respect to $x$ , the derivative of $- 42 x$ with respect to $x$ is $- 42 \frac{d}{d x} [x]$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42 \frac{d}{d x} [x] + \frac{d}{d x} [- 56]$

Step 3.7.2

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

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42 \cdot 1 + \frac{d}{d x} [- 56]$

Step 3.7.3

Multiply $- 42$ by $1$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42 + \frac{d}{d x} [- 56]$

Step 3.8

Differentiate using the Constant Rule.

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Step 3.8.1

Since $- 56$ is constant with respect to $x$ , the derivative of $- 56$ with respect to $x$ is $0$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42 + 0$

Step 3.8.2

Add $336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42$ and $0$ .

$f''' (x) = 336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42$ .

$336 x^{5} - 420 x^{4} - 3000 x^{3} + 660 x^{2} - 720 x - 42$