Calculus Examples

$f (x) = (2 x^{3} + 3 x) (x - 1) (x + 3)$

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 $f (x) = (2 x^{3} + 3 x) (x - 1)$ and $g (x) = x + 3$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(2 x^{3} + 3 x) (x - 1) (1 + 0) + (x + 3) \frac{d}{d x} [(2 x^{3} + 3 x) (x - 1)]$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

Multiply $2 x^{3} + 3 x$ by $1$ .

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

Step 1.3

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 $f (x) = 2 x^{3} + 3 x$ and $g (x) = x - 1$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

$(2 x^{3} + 3 x) (x - 1) + (x + 3) ((2 x^{3} + 3 x) (1 + 0) + (x - 1) \frac{d}{d x} [2 x^{3} + 3 x])$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Add $1$ and $0$ .

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

Step 1.4.4.2

Multiply $2 x^{3} + 3 x$ by $1$ .

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

Step 1.4.5

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

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

Step 1.4.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 1.4.7

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

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

Step 1.4.8

Multiply $3$ by $2$ .

$(2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + \frac{d}{d x} [3 x]))$

Step 1.4.9

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

$(2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3 \frac{d}{d x} [x]))$

Step 1.4.10

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

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

Step 1.4.11

Multiply $3$ by $1$ .

$(2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3))$

Step 1.5

Simplify.

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Step 1.5.1

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

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Step 1.5.1.1

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

$1 ((2 x^{3} + 3 x) (x - 1)) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3))$

Step 1.5.1.2

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

$1 ((2 x^{3} + 3 x) (x - 1)) + 1 ((x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3)))$

Step 1.5.1.3

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

$1 ((2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3)))$

Step 1.5.2

Multiply $(2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3))$ by $1$ .

$(2 x^{3} + 3 x) (x - 1) + (x + 3) (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3))$

Step 1.5.3

Reorder terms.

$(2 x^{3} + 3 x) (x - 1) + (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3)) (x + 3)$

Step 1.5.4

Simplify each term.

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Step 1.5.4.1

Expand $(2 x^{3} + 3 x) (x - 1)$ using the FOIL Method.

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Step 1.5.4.1.1

Apply the distributive property.

$2 x^{3} (x - 1) + 3 x (x - 1) + (2 x^{3} + 3 x + (x - 1) (6 x^{2} + 3)) (x + 3)$

Step 1.5.4.1.2

Apply the distributive property.

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

Step 1.5.4.1.3

Apply the distributive property.

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

Step 1.5.4.2

Simplify each term.

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Step 1.5.4.2.1

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

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Step 1.5.4.2.1.1

Move $x$ .

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

Step 1.5.4.2.1.2

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

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Step 1.5.4.2.1.2.1

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

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

Step 1.5.4.2.1.2.2

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

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

Step 1.5.4.2.1.3

Add $1$ and $3$ .

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

Step 1.5.4.2.2

Multiply $- 1$ by $2$ .

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

Step 1.5.4.2.3

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

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Step 1.5.4.2.3.1

Move $x$ .

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

Step 1.5.4.2.3.2

Multiply $x$ by $x$ .

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

Step 1.5.4.2.4

Multiply $- 1$ by $3$ .

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

Step 1.5.4.3

Simplify each term.

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Step 1.5.4.3.1

Expand $(x - 1) (6 x^{2} + 3)$ using the FOIL Method.

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Step 1.5.4.3.1.1

Apply the distributive property.

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

Step 1.5.4.3.1.2

Apply the distributive property.

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

Step 1.5.4.3.1.3

Apply the distributive property.

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

Step 1.5.4.3.2

Simplify each term.

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Step 1.5.4.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.4.3.2.2

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

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Step 1.5.4.3.2.2.1

Move $x^{2}$ .

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

Step 1.5.4.3.2.2.2

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

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Step 1.5.4.3.2.2.2.1

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

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

Step 1.5.4.3.2.2.2.2

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

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

Step 1.5.4.3.2.2.3

Add $2$ and $1$ .

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

Step 1.5.4.3.2.3

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

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

Step 1.5.4.3.2.4

Multiply $6$ by $- 1$ .

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

Step 1.5.4.3.2.5

Multiply $- 1$ by $3$ .

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

Step 1.5.4.4

Add $2 x^{3}$ and $6 x^{3}$ .

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

Step 1.5.4.5

Add $3 x$ and $3 x$ .

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

Step 1.5.4.6

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

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{3} x + 8 x^{3} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 6 x^{2} x - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7

Simplify each term.

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Step 1.5.4.7.1

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

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Step 1.5.4.7.1.1

Move $x$ .

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

Step 1.5.4.7.1.2

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

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Step 1.5.4.7.1.2.1

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

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

Step 1.5.4.7.1.2.2

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

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

Step 1.5.4.7.1.3

Add $1$ and $3$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 8 x^{3} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 6 x^{2} x - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.2

Multiply $3$ by $8$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x \cdot x + 6 x \cdot 3 - 6 x^{2} x - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.3

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

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Step 1.5.4.7.3.1

Move $x$ .

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

Step 1.5.4.7.3.2

Multiply $x$ by $x$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 6 x \cdot 3 - 6 x^{2} x - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.4

Multiply $3$ by $6$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 x^{2} x - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.5

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

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Step 1.5.4.7.5.1

Move $x$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 (x \cdot x^{2}) - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.5.2

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

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Step 1.5.4.7.5.2.1

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

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 (x^{1} x^{2}) - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.5.2.2

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

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 x^{1 + 2} - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.5.3

Add $1$ and $2$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 x^{3} - 6 x^{2} \cdot 3 - 3 x - 3 \cdot 3$

Step 1.5.4.7.6

Multiply $3$ by $- 6$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 x^{3} - 18 x^{2} - 3 x - 3 \cdot 3$

Step 1.5.4.7.7

Multiply $- 3$ by $3$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 24 x^{3} + 6 x^{2} + 18 x - 6 x^{3} - 18 x^{2} - 3 x - 9$

Step 1.5.4.8

Subtract $6 x^{3}$ from $24 x^{3}$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 18 x^{3} + 6 x^{2} + 18 x - 18 x^{2} - 3 x - 9$

Step 1.5.4.9

Subtract $18 x^{2}$ from $6 x^{2}$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 18 x^{3} - 12 x^{2} + 18 x - 3 x - 9$

Step 1.5.4.10

Subtract $3 x$ from $18 x$ .

$2 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 8 x^{4} + 18 x^{3} - 12 x^{2} + 15 x - 9$

Step 1.5.5

Add $2 x^{4}$ and $8 x^{4}$ .

$10 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 18 x^{3} - 12 x^{2} + 15 x - 9$

Step 1.5.6

Add $- 2 x^{3}$ and $18 x^{3}$ .

$10 x^{4} + 16 x^{3} + 3 x^{2} - 3 x - 12 x^{2} + 15 x - 9$

Step 1.5.7

Subtract $12 x^{2}$ from $3 x^{2}$ .

$10 x^{4} + 16 x^{3} - 9 x^{2} - 3 x + 15 x - 9$

Step 1.5.8

Add $- 3 x$ and $15 x$ .

$f' (x) = 10 x^{4} + 16 x^{3} - 9 x^{2} + 12 x - 9$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $10 x^{4} + 16 x^{3} - 9 x^{2} + 12 x - 9$ with respect to $x$ is $\frac{d}{d x} [10 x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [10 x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.2

Evaluate $\frac{d}{d x} [10 x^{4}]$ .

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Step 2.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{4}$ with respect to $x$ is $10 \frac{d}{d x} [x^{4}]$ .

$10 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.2.2

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

$10 (4 x^{3}) + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.2.3

Multiply $4$ by $10$ .

$40 x^{3} + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.3

Evaluate $\frac{d}{d x} [16 x^{3}]$ .

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Step 2.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3}$ with respect to $x$ is $16 \frac{d}{d x} [x^{3}]$ .

$40 x^{3} + 16 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.3.2

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

$40 x^{3} + 16 (3 x^{2}) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.3.3

Multiply $3$ by $16$ .

$40 x^{3} + 48 x^{2} + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.4

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

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Step 2.4.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$40 x^{3} + 48 x^{2} - 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.4.2

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

$40 x^{3} + 48 x^{2} - 9 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.4.3

Multiply $2$ by $- 9$ .

$40 x^{3} + 48 x^{2} - 18 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 9]$

Step 2.5

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 2.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$40 x^{3} + 48 x^{2} - 18 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

Step 2.5.2

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

$40 x^{3} + 48 x^{2} - 18 x + 12 \cdot 1 + \frac{d}{d x} [- 9]$

Step 2.5.3

Multiply $12$ by $1$ .

$40 x^{3} + 48 x^{2} - 18 x + 12 + \frac{d}{d x} [- 9]$

Step 2.6

Differentiate using the Constant Rule.

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Step 2.6.1

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

$40 x^{3} + 48 x^{2} - 18 x + 12 + 0$

Step 2.6.2

Add $40 x^{3} + 48 x^{2} - 18 x + 12$ and $0$ .

$f'' (x) = 40 x^{3} + 48 x^{2} - 18 x + 12$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $40 x^{3} + 48 x^{2} - 18 x + 12$ with respect to $x$ is $\frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.2

Evaluate $\frac{d}{d x} [40 x^{3}]$ .

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Step 3.2.1

Since $40$ is constant with respect to $x$ , the derivative of $40 x^{3}$ with respect to $x$ is $40 \frac{d}{d x} [x^{3}]$ .

$40 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.2.2

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

$40 (3 x^{2}) + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.2.3

Multiply $3$ by $40$ .

$120 x^{2} + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.3

Evaluate $\frac{d}{d x} [48 x^{2}]$ .

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Step 3.3.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x^{2}$ with respect to $x$ is $48 \frac{d}{d x} [x^{2}]$ .

$120 x^{2} + 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.3.2

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

$120 x^{2} + 48 (2 x) + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.3.3

Multiply $2$ by $48$ .

$120 x^{2} + 96 x + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 3.4

Evaluate $\frac{d}{d x} [- 18 x]$ .

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Step 3.4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$120 x^{2} + 96 x - 18 \frac{d}{d x} [x] + \frac{d}{d x} [12]$

Step 3.4.2

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

$120 x^{2} + 96 x - 18 \cdot 1 + \frac{d}{d x} [12]$

Step 3.4.3

Multiply $- 18$ by $1$ .

$120 x^{2} + 96 x - 18 + \frac{d}{d x} [12]$

Step 3.5

Differentiate using the Constant Rule.

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Step 3.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$120 x^{2} + 96 x - 18 + 0$

Step 3.5.2

Add $120 x^{2} + 96 x - 18$ and $0$ .

$f''' (x) = 120 x^{2} + 96 x - 18$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $120 x^{2} + 96 x - 18$ with respect to $x$ is $\frac{d}{d x} [120 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 18]$ .

$\frac{d}{d x} [120 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 18]$

Step 4.2

Evaluate $\frac{d}{d x} [120 x^{2}]$ .

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Step 4.2.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x^{2}$ with respect to $x$ is $120 \frac{d}{d x} [x^{2}]$ .

$120 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 18]$

Step 4.2.2

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

$120 (2 x) + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 18]$

Step 4.2.3

Multiply $2$ by $120$ .

$240 x + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 18]$

Step 4.3

Evaluate $\frac{d}{d x} [96 x]$ .

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Step 4.3.1

Since $96$ is constant with respect to $x$ , the derivative of $96 x$ with respect to $x$ is $96 \frac{d}{d x} [x]$ .

$240 x + 96 \frac{d}{d x} [x] + \frac{d}{d x} [- 18]$

Step 4.3.2

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

$240 x + 96 \cdot 1 + \frac{d}{d x} [- 18]$

Step 4.3.3

Multiply $96$ by $1$ .

$240 x + 96 + \frac{d}{d x} [- 18]$

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18$ with respect to $x$ is $0$ .

$240 x + 96 + 0$

Step 4.4.2

Add $240 x + 96$ and $0$ .

$f^{4} (x) = 240 x + 96$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $240 x + 96$ .

$240 x + 96$