Calculus Examples

$f (x) = 8 x + \frac{2 x - \frac{2}{3}}{3}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $2$ out of $2 x - \frac{2}{3}$ .

Tap for more steps...

Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

Factor $2$ out of $- \frac{2}{3}$ .

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

Step 1.1.3

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

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

Step 1.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.3

Combine $x$ and $\frac{3}{3}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

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

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

Step 2

Combine $2$ and $\frac{3 x - 1}{3}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{2 (3 x - 1)}{3} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 4.1

Multiply $\frac{2 (3 x - 1)}{3}$ by $\frac{1}{3}$ .

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

Step 4.2

Multiply $3$ by $3$ .

$\frac{d}{d x} [8 x + \frac{2 (3 x - 1)}{9}]$

Step 5

By the Sum Rule, the derivative of $8 x + \frac{2 (3 x - 1)}{9}$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$

Step 6

Evaluate $\frac{d}{d x} [8 x]$ .

Tap for more steps...

Step 6.1

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

$8 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$

Step 6.2

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

$8 \cdot 1 + \frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$

Step 6.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$

Step 7

Evaluate $\frac{d}{d x} [\frac{2 (3 x - 1)}{9}]$ .

Tap for more steps...

Step 7.1

Since $\frac{2}{9}$ is constant with respect to $x$ , the derivative of $\frac{2 (3 x - 1)}{9}$ with respect to $x$ is $\frac{2}{9} \frac{d}{d x} [3 x - 1]$ .

$8 + \frac{2}{9} \frac{d}{d x} [3 x - 1]$

Step 7.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$8 + \frac{2}{9} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 7.3

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

$8 + \frac{2}{9} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 7.4

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

$8 + \frac{2}{9} (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 7.5

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

$8 + \frac{2}{9} (3 \cdot 1 + 0)$

Step 7.6

Multiply $3$ by $1$ .

$8 + \frac{2}{9} (3 + 0)$

Step 7.7

Add $3$ and $0$ .

$8 + \frac{2}{9} \cdot 3$

Step 7.8

Combine $\frac{2}{9}$ and $3$ .

$8 + \frac{2 \cdot 3}{9}$

Step 7.9

Multiply $2$ by $3$ .

$8 + \frac{6}{9}$

Step 7.10

Cancel the common factor of $6$ and $9$ .

Tap for more steps...

Step 7.10.1

Factor $3$ out of $6$ .

$8 + \frac{3 (2)}{9}$

Step 7.10.2

Cancel the common factors.

Tap for more steps...

Step 7.10.2.1

Factor $3$ out of $9$ .

$8 + \frac{3 \cdot 2}{3 \cdot 3}$

Step 7.10.2.2

Cancel the common factor.

$8 + \frac{3 \cdot 2}{3 \cdot 3}$

Step 7.10.2.3

Rewrite the expression.

$8 + \frac{2}{3}$

Step 8

Combine terms.

Tap for more steps...

Step 8.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} + \frac{2}{3}$

Step 8.2

Combine $8$ and $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3} + \frac{2}{3}$

Step 8.3

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 + 2}{3}$

Step 8.4

Simplify the numerator.

Tap for more steps...

Step 8.4.1

Multiply $8$ by $3$ .

$\frac{24 + 2}{3}$

Step 8.4.2

Add $24$ and $2$ .

$\frac{26}{3}$