Calculus Examples

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

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

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

Step 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 $g (x) = x + 2$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Move $3$ to the left of $x - 6$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$3 (x - 6) {(x + 2)}^{2} (1 + 0) + {(x + 2)}^{3} \frac{d}{d x} [x - 6]$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

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

Step 3.5.2

Multiply $3$ by $1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$3 (x - 6) {(x + 2)}^{2} + {(x + 2)}^{3} (1 + 0)$

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $1$ and $0$ .

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

Step 3.9.2

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $- 6$ .

$(3 x - 18) {(x + 2)}^{2} + {(x + 2)}^{3}$

Step 4.3

Factor ${(x + 2)}^{2}$ out of $(3 x - 18) {(x + 2)}^{2} + {(x + 2)}^{3}$ .

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Step 4.3.1

Factor ${(x + 2)}^{2}$ out of $(3 x - 18) {(x + 2)}^{2}$ .

${(x + 2)}^{2} (3 x - 18) + {(x + 2)}^{3}$

Step 4.3.2

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

${(x + 2)}^{2} (3 x - 18) + {(x + 2)}^{2} (x + 2)$

Step 4.3.3

Factor ${(x + 2)}^{2}$ out of ${(x + 2)}^{2} (3 x - 18) + {(x + 2)}^{2} (x + 2)$ .

${(x + 2)}^{2} (3 x - 18 + x + 2)$

Step 4.4

Combine terms.

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Step 4.4.1

Add $3 x$ and $x$ .

${(x + 2)}^{2} (4 x - 18 + 2)$

Step 4.4.2

Add $- 18$ and $2$ .

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

Step 4.5

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

$(x + 2) (x + 2) (4 x - 16)$

Step 4.6

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

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Step 4.6.1

Apply the distributive property.

$(x (x + 2) + 2 (x + 2)) (4 x - 16)$

Step 4.6.2

Apply the distributive property.

$(x \cdot x + x \cdot 2 + 2 (x + 2)) (4 x - 16)$

Step 4.6.3

Apply the distributive property.

$(x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) (4 x - 16)$

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Multiply $x$ by $x$ .

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

Step 4.7.1.2

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

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

Step 4.7.1.3

Multiply $2$ by $2$ .

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

Step 4.7.2

Add $2 x$ and $2 x$ .

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

Step 4.8

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

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

Step 4.9

Simplify each term.

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Step 4.9.1

Rewrite using the commutative property of multiplication.

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

Step 4.9.2

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

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Step 4.9.2.1

Move $x$ .

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

Step 4.9.2.2

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

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Step 4.9.2.2.1

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

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

Step 4.9.2.2.2

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

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

Step 4.9.2.3

Add $1$ and $2$ .

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

Step 4.9.3

Move $- 16$ to the left of $x^{2}$ .

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

Step 4.9.4

Rewrite using the commutative property of multiplication.

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

Step 4.9.5

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

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Step 4.9.5.1

Move $x$ .

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

Step 4.9.5.2

Multiply $x$ by $x$ .

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

Step 4.9.6

Multiply $4$ by $4$ .

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

Step 4.9.7

Multiply $- 16$ by $4$ .

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

Step 4.9.8

Multiply $4$ by $4$ .

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

Step 4.9.9

Multiply $4$ by $- 16$ .

$4 x^{3} - 16 x^{2} + 16 x^{2} - 64 x + 16 x - 64$

Step 4.10

Combine the opposite terms in $4 x^{3} - 16 x^{2} + 16 x^{2} - 64 x + 16 x - 64$ .

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Step 4.10.1

Add $- 16 x^{2}$ and $16 x^{2}$ .

$4 x^{3} + 0 - 64 x + 16 x - 64$

Step 4.10.2

Add $4 x^{3}$ and $0$ .

$4 x^{3} - 64 x + 16 x - 64$

Step 4.11

Add $- 64 x$ and $16 x$ .

$4 x^{3} - 48 x - 64$