Calculus Examples

$(8 - 2 x) (3 - 2 x) x$

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 $f (x) = (8 - 2 x) (3 - 2 x)$ and $g (x) = x$ .

$(8 - 2 x) (3 - 2 x) \frac{d}{d x} [x] + x \frac{d}{d x} [(8 - 2 x) (3 - 2 x)]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$(8 - 2 x) (3 - 2 x) \cdot 1 + x \frac{d}{d x} [(8 - 2 x) (3 - 2 x)]$

Step 2.2

Multiply $8 - 2 x$ by $1$ .

$(8 - 2 x) (3 - 2 x) + x \frac{d}{d x} [(8 - 2 x) (3 - 2 x)]$

Step 3

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 $f (x) = 8 - 2 x$ and $g (x) = 3 - 2 x$ .

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) \frac{d}{d x} [3 - 2 x] + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4

Differentiate.

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Step 4.1

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

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) (\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x]) + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) (0 + \frac{d}{d x} [- 2 x]) + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) \frac{d}{d x} [- 2 x] + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.4

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

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) (- 2 \frac{d}{d x} [x]) + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.5

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

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) (- 2 \cdot 1) + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Multiply $- 2$ by $1$ .

$(8 - 2 x) (3 - 2 x) + x ((8 - 2 x) \cdot - 2 + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.6.2

Move $- 2$ to the left of $8 - 2 x$ .

$(8 - 2 x) (3 - 2 x) + x (- 2 \cdot (8 - 2 x) + (3 - 2 x) \frac{d}{d x} [8 - 2 x])$

Step 4.7

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

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) (\frac{d}{d x} [8] + \frac{d}{d x} [- 2 x]))$

Step 4.8

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) (0 + \frac{d}{d x} [- 2 x]))$

Step 4.9

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) \frac{d}{d x} [- 2 x])$

Step 4.10

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

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) (- 2 \frac{d}{d x} [x]))$

Step 4.11

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

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) (- 2 \cdot 1))$

Step 4.12

Simplify the expression.

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Step 4.12.1

Multiply $- 2$ by $1$ .

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) + (3 - 2 x) \cdot - 2)$

Step 4.12.2

Move $- 2$ to the left of $3 - 2 x$ .

$(8 - 2 x) (3 - 2 x) + x (- 2 (8 - 2 x) - 2 \cdot (3 - 2 x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$(8 - 2 x) (3 - 2 x) + x (- 2 \cdot 8 - 2 (- 2 x) - 2 (3 - 2 x))$

Step 5.2

Apply the distributive property.

$(8 - 2 x) (3 - 2 x) + x (- 2 \cdot 8 - 2 (- 2 x) - 2 \cdot 3 - 2 (- 2 x))$

Step 5.3

Apply the distributive property.

$(8 - 2 x) (3 - 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.4

Apply the distributive property.

$(8 - 2 x) \cdot 3 + (8 - 2 x) (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.5

Apply the distributive property.

$8 \cdot 3 - 2 x \cdot 3 + (8 - 2 x) (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.6

Apply the distributive property.

$8 \cdot 3 - 2 x \cdot 3 + 8 (- 2 x) - 2 x (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7

Combine terms.

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Step 5.7.1

Multiply $8$ by $3$ .

$24 - 2 x \cdot 3 + 8 (- 2 x) - 2 x (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.2

Multiply $3$ by $- 2$ .

$24 - 6 x + 8 (- 2 x) - 2 x (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.3

Multiply $- 2$ by $8$ .

$24 - 6 x - 16 x - 2 x (- 2 x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.4

Multiply $- 2$ by $- 2$ .

$24 - 6 x - 16 x + 4 x \cdot x + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.5

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

$24 - 6 x - 16 x + 4 (x^{1} x) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.6

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

$24 - 6 x - 16 x + 4 (x^{1} x^{1}) + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.7

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

$24 - 6 x - 16 x + 4 x^{1 + 1} + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.8

Add $1$ and $1$ .

$24 - 6 x - 16 x + 4 x^{2} + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.9

Subtract $16 x$ from $- 6 x$ .

$24 - 22 x + 4 x^{2} + x (- 2 \cdot 8) + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.10

Multiply $- 2$ by $8$ .

$24 - 22 x + 4 x^{2} + x \cdot - 16 + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.11

Move $- 16$ to the left of $x$ .

$24 - 22 x + 4 x^{2} - 16 \cdot x + x (- 2 (- 2 x)) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.12

Multiply $- 2$ by $- 2$ .

$24 - 22 x + 4 x^{2} - 16 x + x (4 x) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.13

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

$24 - 22 x + 4 x^{2} - 16 x + 4 (x^{1} x) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.14

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

$24 - 22 x + 4 x^{2} - 16 x + 4 (x^{1} x^{1}) + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.15

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

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{1 + 1} + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.16

Add $1$ and $1$ .

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} + x (- 2 \cdot 3) + x (- 2 (- 2 x))$

Step 5.7.17

Multiply $- 2$ by $3$ .

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} + x \cdot - 6 + x (- 2 (- 2 x))$

Step 5.7.18

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

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 \cdot x + x (- 2 (- 2 x))$

Step 5.7.19

Multiply $- 2$ by $- 2$ .

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 x + x (4 x)$

Step 5.7.20

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

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 x + 4 (x^{1} x)$

Step 5.7.21

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

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 x + 4 (x^{1} x^{1})$

Step 5.7.22

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

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 x + 4 x^{1 + 1}$

Step 5.7.23

Add $1$ and $1$ .

$24 - 22 x + 4 x^{2} - 16 x + 4 x^{2} - 6 x + 4 x^{2}$

Step 5.7.24

Subtract $6 x$ from $- 16 x$ .

$24 - 22 x + 4 x^{2} + 4 x^{2} - 22 x + 4 x^{2}$

Step 5.7.25

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

$24 - 22 x + 4 x^{2} + 8 x^{2} - 22 x$

Step 5.7.26

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

$24 - 22 x + 12 x^{2} - 22 x$

Step 5.7.27

Subtract $22 x$ from $- 22 x$ .

$24 + 12 x^{2} - 44 x$

Step 5.8

Reorder terms.

$12 x^{2} - 44 x + 24$