Calculus Examples

$\frac{x^{3} - 8 x}{4}$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{3} - 8 x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{3} - 8 x]$ .

$\frac{1}{4} \frac{d}{d x} [x^{3} - 8 x]$

Step 2

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

$\frac{1}{4} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x])$

Step 3

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

$\frac{1}{4} (3 x^{2} + \frac{d}{d x} [- 8 x])$

Step 4

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

$\frac{1}{4} (3 x^{2} - 8 \frac{d}{d x} [x])$

Step 5

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

$\frac{1}{4} (3 x^{2} - 8 \cdot 1)$

Step 6

Multiply $- 8$ by $1$ .

$\frac{1}{4} (3 x^{2} - 8)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

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

Step 7.2

Combine terms.

Tap for more steps...

Step 7.2.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4} x^{2} + \frac{1}{4} \cdot - 8$

Step 7.2.2

Combine $\frac{3}{4}$ and $x^{2}$ .

$\frac{3 x^{2}}{4} + \frac{1}{4} \cdot - 8$

Step 7.2.3

Combine $\frac{1}{4}$ and $- 8$ .

$\frac{3 x^{2}}{4} + \frac{- 8}{4}$

Step 7.2.4

Cancel the common factor of $- 8$ and $4$ .

Tap for more steps...

Step 7.2.4.1

Factor $4$ out of $- 8$ .

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

Step 7.2.4.2

Cancel the common factors.

Tap for more steps...

Step 7.2.4.2.1

Factor $4$ out of $4$ .

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

Step 7.2.4.2.2

Cancel the common factor.

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

Step 7.2.4.2.3

Rewrite the expression.

$\frac{3 x^{2}}{4} + \frac{- 2}{1}$

Step 7.2.4.2.4

Divide $- 2$ by $1$ .

$\frac{3 x^{2}}{4} - 2$