Calculus Examples

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

Step 1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x {(x + 2)}^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x {(x + 2)}^{3}]$ .

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

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

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

Step 3

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 $g (x) = x + 2$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x + 2$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $3$ by $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $3$ by $- 4$ .

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

Step 5.3

Factor $4 {(x + 2)}^{2}$ out of $- 12 x {(x + 2)}^{2} - 4 {(x + 2)}^{3}$ .

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Step 5.3.1

Factor $4 {(x + 2)}^{2}$ out of $- 12 x {(x + 2)}^{2}$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

$4 ((x + 2) (x + 2)) (- 3 x - (x + 2))$

Step 5.5

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

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Step 5.5.1

Apply the distributive property.

$4 (x (x + 2) + 2 (x + 2)) (- 3 x - (x + 2))$

Step 5.5.2

Apply the distributive property.

$4 (x \cdot x + x \cdot 2 + 2 (x + 2)) (- 3 x - (x + 2))$

Step 5.5.3

Apply the distributive property.

$4 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) (- 3 x - (x + 2))$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $x$ by $x$ .

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

Step 5.6.1.2

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

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

Step 5.6.1.3

Multiply $2$ by $2$ .

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

Step 5.6.2

Add $2 x$ and $2 x$ .

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

Step 5.7

Apply the distributive property.

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

Step 5.8

Simplify.

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Step 5.8.1

Multiply $4$ by $4$ .

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

Step 5.8.2

Multiply $4$ by $4$ .

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

Step 5.9

Simplify each term.

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Step 5.9.1

Apply the distributive property.

$(4 x^{2} + 16 x + 16) (- 3 x - x - 1 \cdot 2)$

Step 5.9.2

Multiply $- 1$ by $2$ .

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

Step 5.10

Subtract $x$ from $- 3 x$ .

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

Step 5.11

Expand $(4 x^{2} + 16 x + 16) (- 4 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.12

Simplify each term.

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Step 5.12.1

Rewrite using the commutative property of multiplication.

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

Step 5.12.2

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

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Step 5.12.2.1

Move $x$ .

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

Step 5.12.2.2

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

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Step 5.12.2.2.1

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

$4 \cdot - 4 (x^{1} x^{2}) + 4 x^{2} \cdot - 2 + 16 x (- 4 x) + 16 x \cdot - 2 + 16 (- 4 x) + 16 \cdot - 2$

Step 5.12.2.2.2

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

$4 \cdot - 4 x^{1 + 2} + 4 x^{2} \cdot - 2 + 16 x (- 4 x) + 16 x \cdot - 2 + 16 (- 4 x) + 16 \cdot - 2$

Step 5.12.2.3

Add $1$ and $2$ .

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

Step 5.12.3

Multiply $4$ by $- 4$ .

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

Step 5.12.4

Multiply $- 2$ by $4$ .

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

Step 5.12.5

Rewrite using the commutative property of multiplication.

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

Step 5.12.6

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

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Step 5.12.6.1

Move $x$ .

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

Step 5.12.6.2

Multiply $x$ by $x$ .

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

Step 5.12.7

Multiply $16$ by $- 4$ .

$- 16 x^{3} - 8 x^{2} - 64 x^{2} + 16 x \cdot - 2 + 16 (- 4 x) + 16 \cdot - 2$

Step 5.12.8

Multiply $- 2$ by $16$ .

$- 16 x^{3} - 8 x^{2} - 64 x^{2} - 32 x + 16 (- 4 x) + 16 \cdot - 2$

Step 5.12.9

Multiply $- 4$ by $16$ .

$- 16 x^{3} - 8 x^{2} - 64 x^{2} - 32 x - 64 x + 16 \cdot - 2$

Step 5.12.10

Multiply $16$ by $- 2$ .

$- 16 x^{3} - 8 x^{2} - 64 x^{2} - 32 x - 64 x - 32$

Step 5.13

Subtract $64 x^{2}$ from $- 8 x^{2}$ .

$- 16 x^{3} - 72 x^{2} - 32 x - 64 x - 32$

Step 5.14

Subtract $64 x$ from $- 32 x$ .

$- 16 x^{3} - 72 x^{2} - 96 x - 32$