Calculus Examples

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

Step 1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$\frac{d}{d x} [(x - 4) (x - 4) (4 x - 4)]$

Step 2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(x (x - 4) - 4 (x - 4)) (4 x - 4)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [(x \cdot x + x \cdot - 4 - 4 (x - 4)) (4 x - 4)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 4$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 4$ by $- 4$ .

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

Step 3.2

Subtract $4 x$ from $- 4 x$ .

$\frac{d}{d x} [(x^{2} - 8 x + 16) (4 x - 4)]$

Step 4

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 $f (x) = x^{2} - 8 x + 16$ and $g (x) = 4 x - 4$ .

$(x^{2} - 8 x + 16) \frac{d}{d x} [4 x - 4] + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $4 x - 4$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]$ .

$(x^{2} - 8 x + 16) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.2

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

$(x^{2} - 8 x + 16) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.3

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

$(x^{2} - 8 x + 16) (4 \cdot 1 + \frac{d}{d x} [- 4]) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.4

Multiply $4$ by $1$ .

$(x^{2} - 8 x + 16) (4 + \frac{d}{d x} [- 4]) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$(x^{2} - 8 x + 16) (4 + 0) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $4$ and $0$ .

$(x^{2} - 8 x + 16) \cdot 4 + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.6.2

Move $4$ to the left of $x^{2} - 8 x + 16$ .

$4 \cdot (x^{2} - 8 x + 16) + (4 x - 4) \frac{d}{d x} [x^{2} - 8 x + 16]$

Step 5.7

By the Sum Rule, the derivative of $x^{2} - 8 x + 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16]$ .

$4 (x^{2} - 8 x + 16) + (4 x - 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16])$

Step 5.8

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

$4 (x^{2} - 8 x + 16) + (4 x - 4) (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16])$

Step 5.9

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$4 (x^{2} - 8 x + 16) + (4 x - 4) (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [16])$

Step 5.10

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

$4 (x^{2} - 8 x + 16) + (4 x - 4) (2 x - 8 \cdot 1 + \frac{d}{d x} [16])$

Step 5.11

Multiply $- 8$ by $1$ .

$4 (x^{2} - 8 x + 16) + (4 x - 4) (2 x - 8 + \frac{d}{d x} [16])$

Step 5.12

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$4 (x^{2} - 8 x + 16) + (4 x - 4) (2 x - 8 + 0)$

Step 5.13

Add $2 x - 8$ and $0$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Combine terms.

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Step 6.5.1

Multiply $- 8$ by $4$ .

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

Step 6.5.2

Multiply $4$ by $16$ .

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

Step 6.5.3

Multiply $2$ by $4$ .

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

Step 6.5.4

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

$4 x^{2} - 32 x + 64 + 8 (x^{1} x) - 4 (2 x) + 4 x \cdot - 8 - 4 \cdot - 8$

Step 6.5.5

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

$4 x^{2} - 32 x + 64 + 8 (x^{1} x^{1}) - 4 (2 x) + 4 x \cdot - 8 - 4 \cdot - 8$

Step 6.5.6

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

$4 x^{2} - 32 x + 64 + 8 x^{1 + 1} - 4 (2 x) + 4 x \cdot - 8 - 4 \cdot - 8$

Step 6.5.7

Add $1$ and $1$ .

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

Step 6.5.8

Multiply $2$ by $- 4$ .

$4 x^{2} - 32 x + 64 + 8 x^{2} - 8 x + 4 x \cdot - 8 - 4 \cdot - 8$

Step 6.5.9

Multiply $- 8$ by $4$ .

$4 x^{2} - 32 x + 64 + 8 x^{2} - 8 x - 32 x - 4 \cdot - 8$

Step 6.5.10

Multiply $- 4$ by $- 8$ .

$4 x^{2} - 32 x + 64 + 8 x^{2} - 8 x - 32 x + 32$

Step 6.5.11

Subtract $32 x$ from $- 8 x$ .

$4 x^{2} - 32 x + 64 + 8 x^{2} - 40 x + 32$

Step 6.5.12

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

$12 x^{2} - 32 x + 64 - 40 x + 32$

Step 6.5.13

Subtract $40 x$ from $- 32 x$ .

$12 x^{2} - 72 x + 64 + 32$

Step 6.5.14

Add $64$ and $32$ .

$12 x^{2} - 72 x + 96$