Calculus Examples

$\frac{x^{4}}{6} - 3 x$

Step 1

By the Sum Rule, the derivative of $\frac{x^{4}}{6} - 3 x$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{6}] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [\frac{x^{4}}{6}] + \frac{d}{d x} [- 3 x]$

Step 2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{6}]$ .

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Step 2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{6} (4 x^{3}) + \frac{d}{d x} [- 3 x]$

Step 2.3

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} x^{3} + \frac{d}{d x} [- 3 x]$

Step 2.4

Combine $\frac{4}{6}$ and $x^{3}$ .

$\frac{4 x^{3}}{6} + \frac{d}{d x} [- 3 x]$

Step 2.5

Cancel the common factor of $4$ and $6$ .

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Step 2.5.1

Factor $2$ out of $4 x^{3}$ .

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

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $6$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 3.1

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

$\frac{2 x^{3}}{3} - 3 \frac{d}{d x} [x]$

Step 3.2

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

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

Step 3.3

Multiply $- 3$ by $1$ .

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