Calculus Examples

$y = 2 {(x^{2} + 2 x)}^{3}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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 $g (x) = x^{2} + 2 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 2 x$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} + 2 x$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $3$ by $2$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Multiply $2$ by $1$ .

$f' (x) = 6 {(x^{2} + 2 x)}^{2} (2 x + 2)$

Step 2

Find the second derivative.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 {(x^{2} + 2 x)}^{2} (2 x + 2)$ with respect to $x$ is $6 \frac{d}{d x} [{(x^{2} + 2 x)}^{2} (2 x + 2)]$ .

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

Step 2.2

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

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $2$ by $1$ .

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

Step 2.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$6 ({(x^{2} + 2 x)}^{2} (2 + 0) + (2 x + 2) \frac{d}{d x} [{(x^{2} + 2 x)}^{2}])$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $2$ and $0$ .

$6 ({(x^{2} + 2 x)}^{2} \cdot 2 + (2 x + 2) \frac{d}{d x} [{(x^{2} + 2 x)}^{2}])$

Step 2.3.6.2

Move $2$ to the left of ${(x^{2} + 2 x)}^{2}$ .

$6 (2 \cdot {(x^{2} + 2 x)}^{2} + (2 x + 2) \frac{d}{d x} [{(x^{2} + 2 x)}^{2}])$

Step 2.4

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

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 2 x$ .

$6 (2 {(x^{2} + 2 x)}^{2} + (2 x + 2) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 2 x]))$

Step 2.4.2

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

$6 (2 {(x^{2} + 2 x)}^{2} + (2 x + 2) (2 u \frac{d}{d x} [x^{2} + 2 x]))$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 2 x$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

Move $2$ to the left of $2 x + 2$ .

$6 (2 {(x^{2} + 2 x)}^{2} + 2 \cdot (2 x + 2) (x^{2} + 2 x) \frac{d}{d x} [x^{2} + 2 x])$

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

Multiply $2$ by $1$ .

$6 (2 {(x^{2} + 2 x)}^{2} + 2 (2 x + 2) (x^{2} + 2 x) (2 x + 2))$

Step 2.6

Raise $2 x + 2$ to the power of $1$ .

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

Step 2.7

Raise $2 x + 2$ to the power of $1$ .

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

Step 2.8

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

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

Step 2.9

Add $1$ and $1$ .

$6 (2 {(x^{2} + 2 x)}^{2} + 2 {(2 x + 2)}^{2} (x^{2} + 2 x))$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$6 (2 {(x^{2} + 2 x)}^{2}) + 6 (2 {(2 x + 2)}^{2} (x^{2} + 2 x))$

Step 2.10.2

Multiply $2$ by $6$ .

$12 {(x^{2} + 2 x)}^{2} + 6 (2 {(2 x + 2)}^{2} (x^{2} + 2 x))$

Step 2.10.3

Multiply $2$ by $6$ .

$12 {(x^{2} + 2 x)}^{2} + 12 ({(2 x + 2)}^{2} (x^{2} + 2 x))$

Step 2.10.4

Factor $12 (x^{2} + 2 x)$ out of $12 {(x^{2} + 2 x)}^{2} + 12 {(2 x + 2)}^{2} (x^{2} + 2 x)$ .

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Step 2.10.4.1

Factor $12 (x^{2} + 2 x)$ out of $12 {(x^{2} + 2 x)}^{2}$ .

$12 (x^{2} + 2 x) (x^{2} + 2 x) + 12 {(2 x + 2)}^{2} (x^{2} + 2 x)$

Step 2.10.4.2

Factor $12 (x^{2} + 2 x)$ out of $12 {(2 x + 2)}^{2} (x^{2} + 2 x)$ .

$12 (x^{2} + 2 x) (x^{2} + 2 x) + 12 (x^{2} + 2 x) ({(2 x + 2)}^{2})$

Step 2.10.4.3

Factor $12 (x^{2} + 2 x)$ out of $12 (x^{2} + 2 x) (x^{2} + 2 x) + 12 (x^{2} + 2 x) ({(2 x + 2)}^{2})$ .

$12 (x^{2} + 2 x) (x^{2} + 2 x + {(2 x + 2)}^{2})$

Step 2.10.5

Apply the distributive property.

$(12 x^{2} + 12 (2 x)) (x^{2} + 2 x + {(2 x + 2)}^{2})$

Step 2.10.6

Multiply $2$ by $12$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + {(2 x + 2)}^{2})$

Step 2.10.7

Simplify each term.

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Step 2.10.7.1

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

$(12 x^{2} + 24 x) (x^{2} + 2 x + (2 x + 2) (2 x + 2))$

Step 2.10.7.2

Expand $(2 x + 2) (2 x + 2)$ using the FOIL Method.

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Step 2.10.7.2.1

Apply the distributive property.

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 x (2 x + 2) + 2 (2 x + 2))$

Step 2.10.7.2.2

Apply the distributive property.

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 x (2 x) + 2 x \cdot 2 + 2 (2 x + 2))$

Step 2.10.7.2.3

Apply the distributive property.

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 x (2 x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3

Simplify and combine like terms.

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Step 2.10.7.3.1

Simplify each term.

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Step 2.10.7.3.1.1

Rewrite using the commutative property of multiplication.

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 \cdot 2 x \cdot x + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3.1.2

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

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Step 2.10.7.3.1.2.1

Move $x$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 \cdot 2 (x \cdot x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3.1.2.2

Multiply $x$ by $x$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 2 \cdot 2 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3.1.3

Multiply $2$ by $2$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 4 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3.1.4

Multiply $2$ by $2$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot 2)$

Step 2.10.7.3.1.5

Multiply $2$ by $2$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 4 x^{2} + 4 x + 4 x + 2 \cdot 2)$

Step 2.10.7.3.1.6

Multiply $2$ by $2$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 4 x^{2} + 4 x + 4 x + 4)$

Step 2.10.7.3.2

Add $4 x$ and $4 x$ .

$(12 x^{2} + 24 x) (x^{2} + 2 x + 4 x^{2} + 8 x + 4)$

Step 2.10.8

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

$(12 x^{2} + 24 x) (5 x^{2} + 2 x + 8 x + 4)$

Step 2.10.9

Add $2 x$ and $8 x$ .

$(12 x^{2} + 24 x) (5 x^{2} + 10 x + 4)$

Step 2.10.10

Expand $(12 x^{2} + 24 x) (5 x^{2} + 10 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$12 x^{2} (5 x^{2}) + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11

Simplify each term.

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Step 2.10.11.1

Rewrite using the commutative property of multiplication.

$12 \cdot 5 x^{2} x^{2} + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.2

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

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Step 2.10.11.2.1

Move $x^{2}$ .

$12 \cdot 5 (x^{2} x^{2}) + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.2.2

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

$12 \cdot 5 x^{2 + 2} + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.2.3

Add $2$ and $2$ .

$12 \cdot 5 x^{4} + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.3

Multiply $12$ by $5$ .

$60 x^{4} + 12 x^{2} (10 x) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.4

Rewrite using the commutative property of multiplication.

$60 x^{4} + 12 \cdot 10 x^{2} x + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.5

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

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Step 2.10.11.5.1

Move $x$ .

$60 x^{4} + 12 \cdot 10 (x \cdot x^{2}) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.5.2

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

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Step 2.10.11.5.2.1

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

$60 x^{4} + 12 \cdot 10 (x^{1} x^{2}) + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.5.2.2

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

$60 x^{4} + 12 \cdot 10 x^{1 + 2} + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.5.3

Add $1$ and $2$ .

$60 x^{4} + 12 \cdot 10 x^{3} + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.6

Multiply $12$ by $10$ .

$60 x^{4} + 120 x^{3} + 12 x^{2} \cdot 4 + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.7

Multiply $4$ by $12$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 x (5 x^{2}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.8

Rewrite using the commutative property of multiplication.

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 \cdot 5 x \cdot x^{2} + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.9

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

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Step 2.10.11.9.1

Move $x^{2}$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 \cdot 5 (x^{2} x) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.9.2

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

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Step 2.10.11.9.2.1

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

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 \cdot 5 (x^{2} x^{1}) + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.9.2.2

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

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 \cdot 5 x^{2 + 1} + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.9.3

Add $2$ and $1$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 24 \cdot 5 x^{3} + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.10

Multiply $24$ by $5$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 24 x (10 x) + 24 x \cdot 4$

Step 2.10.11.11

Rewrite using the commutative property of multiplication.

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 24 \cdot 10 x \cdot x + 24 x \cdot 4$

Step 2.10.11.12

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

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Step 2.10.11.12.1

Move $x$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 24 \cdot 10 (x \cdot x) + 24 x \cdot 4$

Step 2.10.11.12.2

Multiply $x$ by $x$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 24 \cdot 10 x^{2} + 24 x \cdot 4$

Step 2.10.11.13

Multiply $24$ by $10$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 240 x^{2} + 24 x \cdot 4$

Step 2.10.11.14

Multiply $4$ by $24$ .

$60 x^{4} + 120 x^{3} + 48 x^{2} + 120 x^{3} + 240 x^{2} + 96 x$

Step 2.10.12

Add $120 x^{3}$ and $120 x^{3}$ .

$60 x^{4} + 240 x^{3} + 48 x^{2} + 240 x^{2} + 96 x$

Step 2.10.13

Add $48 x^{2}$ and $240 x^{2}$ .

$f'' (x) = 60 x^{4} + 240 x^{3} + 288 x^{2} + 96 x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $60 x^{4} + 240 x^{3} + 288 x^{2} + 96 x$ with respect to $x$ is $\frac{d}{d x} [60 x^{4}] + \frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$ .

$\frac{d}{d x} [60 x^{4}] + \frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.2

Evaluate $\frac{d}{d x} [60 x^{4}]$ .

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Step 3.2.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{4}$ with respect to $x$ is $60 \frac{d}{d x} [x^{4}]$ .

$60 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.2.2

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

$60 (4 x^{3}) + \frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.2.3

Multiply $4$ by $60$ .

$240 x^{3} + \frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.3

Evaluate $\frac{d}{d x} [240 x^{3}]$ .

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Step 3.3.1

Since $240$ is constant with respect to $x$ , the derivative of $240 x^{3}$ with respect to $x$ is $240 \frac{d}{d x} [x^{3}]$ .

$240 x^{3} + 240 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.3.2

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

$240 x^{3} + 240 (3 x^{2}) + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.3.3

Multiply $3$ by $240$ .

$240 x^{3} + 720 x^{2} + \frac{d}{d x} [288 x^{2}] + \frac{d}{d x} [96 x]$

Step 3.4

Evaluate $\frac{d}{d x} [288 x^{2}]$ .

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Step 3.4.1

Since $288$ is constant with respect to $x$ , the derivative of $288 x^{2}$ with respect to $x$ is $288 \frac{d}{d x} [x^{2}]$ .

$240 x^{3} + 720 x^{2} + 288 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [96 x]$

Step 3.4.2

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

$240 x^{3} + 720 x^{2} + 288 (2 x) + \frac{d}{d x} [96 x]$

Step 3.4.3

Multiply $2$ by $288$ .

$240 x^{3} + 720 x^{2} + 576 x + \frac{d}{d x} [96 x]$

Step 3.5

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

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Step 3.5.1

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

$240 x^{3} + 720 x^{2} + 576 x + 96 \frac{d}{d x} [x]$

Step 3.5.2

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

$240 x^{3} + 720 x^{2} + 576 x + 96 \cdot 1$

Step 3.5.3

Multiply $96$ by $1$ .

$f''' (x) = 240 x^{3} + 720 x^{2} + 576 x + 96$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $240 x^{3} + 720 x^{2} + 576 x + 96$ with respect to $x$ is $\frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [720 x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$ .

$\frac{d}{d x} [240 x^{3}] + \frac{d}{d x} [720 x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.2

Evaluate $\frac{d}{d x} [240 x^{3}]$ .

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Step 4.2.1

Since $240$ is constant with respect to $x$ , the derivative of $240 x^{3}$ with respect to $x$ is $240 \frac{d}{d x} [x^{3}]$ .

$240 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [720 x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.2.2

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

$240 (3 x^{2}) + \frac{d}{d x} [720 x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.2.3

Multiply $3$ by $240$ .

$720 x^{2} + \frac{d}{d x} [720 x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.3

Evaluate $\frac{d}{d x} [720 x^{2}]$ .

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Step 4.3.1

Since $720$ is constant with respect to $x$ , the derivative of $720 x^{2}$ with respect to $x$ is $720 \frac{d}{d x} [x^{2}]$ .

$720 x^{2} + 720 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.3.2

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

$720 x^{2} + 720 (2 x) + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.3.3

Multiply $2$ by $720$ .

$720 x^{2} + 1440 x + \frac{d}{d x} [576 x] + \frac{d}{d x} [96]$

Step 4.4

Evaluate $\frac{d}{d x} [576 x]$ .

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Step 4.4.1

Since $576$ is constant with respect to $x$ , the derivative of $576 x$ with respect to $x$ is $576 \frac{d}{d x} [x]$ .

$720 x^{2} + 1440 x + 576 \frac{d}{d x} [x] + \frac{d}{d x} [96]$

Step 4.4.2

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

$720 x^{2} + 1440 x + 576 \cdot 1 + \frac{d}{d x} [96]$

Step 4.4.3

Multiply $576$ by $1$ .

$720 x^{2} + 1440 x + 576 + \frac{d}{d x} [96]$

Step 4.5

Differentiate using the Constant Rule.

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Step 4.5.1

Since $96$ is constant with respect to $x$ , the derivative of $96$ with respect to $x$ is $0$ .

$720 x^{2} + 1440 x + 576 + 0$

Step 4.5.2

Add $720 x^{2} + 1440 x + 576$ and $0$ .

$f^{4} (x) = 720 x^{2} + 1440 x + 576$