Calculus Examples

$x^{4} {(x - 1)}^{3}$

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 $f (x) = x^{4}$ and $g (x) = {(x - 1)}^{3}$ .

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

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 $g (x) = x - 1$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

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

Step 1.3.4.2

Multiply $3$ by $1$ .

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

Step 1.3.5

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

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

Step 1.3.6

Move $4$ to the left of ${(x - 1)}^{3}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

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

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Step 1.4.1.1

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

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

Step 1.4.1.2

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

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

Step 1.4.1.3

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.4.4.1

Apply the distributive property.

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

Step 1.4.4.2

Apply the distributive property.

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

Step 1.4.4.3

Apply the distributive property.

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

Step 1.4.5

Simplify and combine like terms.

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Step 1.4.5.1

Simplify each term.

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Step 1.4.5.1.1

Multiply $x$ by $x$ .

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

Step 1.4.5.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.4.5.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.4.5.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.4.5.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.4.5.2

Subtract $x$ from $- x$ .

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

Step 1.4.6

Apply the distributive property.

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

Step 1.4.7

Simplify.

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Step 1.4.7.1

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

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Step 1.4.7.1.1

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

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

Step 1.4.7.1.2

Add $3$ and $2$ .

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

Step 1.4.7.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.7.3

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

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

Step 1.4.8

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

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Step 1.4.8.1

Move $x$ .

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

Step 1.4.8.2

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

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Step 1.4.8.2.1

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

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

Step 1.4.8.2.2

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

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

Step 1.4.8.3

Add $1$ and $3$ .

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

Step 1.4.9

Simplify each term.

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Step 1.4.9.1

Apply the distributive property.

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

Step 1.4.9.2

Multiply $4$ by $- 1$ .

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

Step 1.4.10

Add $3 x$ and $4 x$ .

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

Step 1.4.11

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

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

Step 1.4.12

Simplify each term.

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Step 1.4.12.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.12.2

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

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Step 1.4.12.2.1

Move $x$ .

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

Step 1.4.12.2.2

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

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Step 1.4.12.2.2.1

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

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

Step 1.4.12.2.2.2

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

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

Step 1.4.12.2.3

Add $1$ and $5$ .

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

Step 1.4.12.3

Move $- 4$ to the left of $x^{5}$ .

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

Step 1.4.12.4

Rewrite using the commutative property of multiplication.

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

Step 1.4.12.5

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

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Step 1.4.12.5.1

Move $x$ .

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

Step 1.4.12.5.2

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

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Step 1.4.12.5.2.1

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

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

Step 1.4.12.5.2.2

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

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

Step 1.4.12.5.3

Add $1$ and $4$ .

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

Step 1.4.12.6

Multiply $- 2$ by $7$ .

$7 x^{6} - 4 x^{5} - 14 x^{5} - 2 x^{4} \cdot - 4 + x^{3} (7 x) + x^{3} \cdot - 4$

Step 1.4.12.7

Multiply $- 4$ by $- 2$ .

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + x^{3} (7 x) + x^{3} \cdot - 4$

Step 1.4.12.8

Rewrite using the commutative property of multiplication.

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 x^{3} x + x^{3} \cdot - 4$

Step 1.4.12.9

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

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Step 1.4.12.9.1

Move $x$ .

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 (x \cdot x^{3}) + x^{3} \cdot - 4$

Step 1.4.12.9.2

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

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Step 1.4.12.9.2.1

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

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 (x^{1} x^{3}) + x^{3} \cdot - 4$

Step 1.4.12.9.2.2

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

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 x^{1 + 3} + x^{3} \cdot - 4$

Step 1.4.12.9.3

Add $1$ and $3$ .

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 x^{4} + x^{3} \cdot - 4$

Step 1.4.12.10

Move $- 4$ to the left of $x^{3}$ .

$7 x^{6} - 4 x^{5} - 14 x^{5} + 8 x^{4} + 7 x^{4} - 4 x^{3}$

Step 1.4.13

Subtract $14 x^{5}$ from $- 4 x^{5}$ .

$7 x^{6} - 18 x^{5} + 8 x^{4} + 7 x^{4} - 4 x^{3}$

Step 1.4.14

Add $8 x^{4}$ and $7 x^{4}$ .

$f' (x) = 7 x^{6} - 18 x^{5} + 15 x^{4} - 4 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $7 x^{6} - 18 x^{5} + 15 x^{4} - 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [7 x^{6}] + \frac{d}{d x} [- 18 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$ .

$\frac{d}{d x} [7 x^{6}] + \frac{d}{d x} [- 18 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [7 x^{6}]$ .

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Step 2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{6}$ with respect to $x$ is $7 \frac{d}{d x} [x^{6}]$ .

$7 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 18 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.2.2

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

$7 (6 x^{5}) + \frac{d}{d x} [- 18 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.2.3

Multiply $6$ by $7$ .

$42 x^{5} + \frac{d}{d x} [- 18 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 18 x^{5}]$ .

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Step 2.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{5}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{5}]$ .

$42 x^{5} - 18 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.3.2

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

$42 x^{5} - 18 (5 x^{4}) + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.3.3

Multiply $5$ by $- 18$ .

$42 x^{5} - 90 x^{4} + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.4

Evaluate $\frac{d}{d x} [15 x^{4}]$ .

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Step 2.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{4}$ with respect to $x$ is $15 \frac{d}{d x} [x^{4}]$ .

$42 x^{5} - 90 x^{4} + 15 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.4.2

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

$42 x^{5} - 90 x^{4} + 15 (4 x^{3}) + \frac{d}{d x} [- 4 x^{3}]$

Step 2.4.3

Multiply $4$ by $15$ .

$42 x^{5} - 90 x^{4} + 60 x^{3} + \frac{d}{d x} [- 4 x^{3}]$

Step 2.5

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 2.5.1

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

$42 x^{5} - 90 x^{4} + 60 x^{3} - 4 \frac{d}{d x} [x^{3}]$

Step 2.5.2

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

$42 x^{5} - 90 x^{4} + 60 x^{3} - 4 (3 x^{2})$

Step 2.5.3

Multiply $3$ by $- 4$ .

$f'' (x) = 42 x^{5} - 90 x^{4} + 60 x^{3} - 12 x^{2}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $42 x^{5} - 90 x^{4} + 60 x^{3} - 12 x^{2}$ with respect to $x$ is $\frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 90 x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$ .

$\frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 90 x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [42 x^{5}]$ .

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Step 3.2.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{5}$ with respect to $x$ is $42 \frac{d}{d x} [x^{5}]$ .

$42 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 90 x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.2.2

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

$42 (5 x^{4}) + \frac{d}{d x} [- 90 x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.2.3

Multiply $5$ by $42$ .

$210 x^{4} + \frac{d}{d x} [- 90 x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 90 x^{4}]$ .

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Step 3.3.1

Since $- 90$ is constant with respect to $x$ , the derivative of $- 90 x^{4}$ with respect to $x$ is $- 90 \frac{d}{d x} [x^{4}]$ .

$210 x^{4} - 90 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.3.2

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

$210 x^{4} - 90 (4 x^{3}) + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.3.3

Multiply $4$ by $- 90$ .

$210 x^{4} - 360 x^{3} + \frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.4

Evaluate $\frac{d}{d x} [60 x^{3}]$ .

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Step 3.4.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{3}$ with respect to $x$ is $60 \frac{d}{d x} [x^{3}]$ .

$210 x^{4} - 360 x^{3} + 60 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 3.4.2

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

$210 x^{4} - 360 x^{3} + 60 (3 x^{2}) + \frac{d}{d x} [- 12 x^{2}]$

Step 3.4.3

Multiply $3$ by $60$ .

$210 x^{4} - 360 x^{3} + 180 x^{2} + \frac{d}{d x} [- 12 x^{2}]$

Step 3.5

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 3.5.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$210 x^{4} - 360 x^{3} + 180 x^{2} - 12 \frac{d}{d x} [x^{2}]$

Step 3.5.2

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

$210 x^{4} - 360 x^{3} + 180 x^{2} - 12 (2 x)$

Step 3.5.3

Multiply $2$ by $- 12$ .

$f''' (x) = 210 x^{4} - 360 x^{3} + 180 x^{2} - 24 x$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $210 x^{4} - 360 x^{3} + 180 x^{2} - 24 x$ with respect to $x$ is $\frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 360 x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$ .

$\frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 360 x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.2

Evaluate $\frac{d}{d x} [210 x^{4}]$ .

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Step 4.2.1

Since $210$ is constant with respect to $x$ , the derivative of $210 x^{4}$ with respect to $x$ is $210 \frac{d}{d x} [x^{4}]$ .

$210 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 360 x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.2.2

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

$210 (4 x^{3}) + \frac{d}{d x} [- 360 x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.2.3

Multiply $4$ by $210$ .

$840 x^{3} + \frac{d}{d x} [- 360 x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.3

Evaluate $\frac{d}{d x} [- 360 x^{3}]$ .

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Step 4.3.1

Since $- 360$ is constant with respect to $x$ , the derivative of $- 360 x^{3}$ with respect to $x$ is $- 360 \frac{d}{d x} [x^{3}]$ .

$840 x^{3} - 360 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.3.2

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

$840 x^{3} - 360 (3 x^{2}) + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.3.3

Multiply $3$ by $- 360$ .

$840 x^{3} - 1080 x^{2} + \frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.4

Evaluate $\frac{d}{d x} [180 x^{2}]$ .

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Step 4.4.1

Since $180$ is constant with respect to $x$ , the derivative of $180 x^{2}$ with respect to $x$ is $180 \frac{d}{d x} [x^{2}]$ .

$840 x^{3} - 1080 x^{2} + 180 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x]$

Step 4.4.2

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

$840 x^{3} - 1080 x^{2} + 180 (2 x) + \frac{d}{d x} [- 24 x]$

Step 4.4.3

Multiply $2$ by $180$ .

$840 x^{3} - 1080 x^{2} + 360 x + \frac{d}{d x} [- 24 x]$

Step 4.5

Evaluate $\frac{d}{d x} [- 24 x]$ .

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Step 4.5.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$840 x^{3} - 1080 x^{2} + 360 x - 24 \frac{d}{d x} [x]$

Step 4.5.2

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

$840 x^{3} - 1080 x^{2} + 360 x - 24 \cdot 1$

Step 4.5.3

Multiply $- 24$ by $1$ .

$f^{4} (x) = 840 x^{3} - 1080 x^{2} + 360 x - 24$