Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{x^{3}}{3} + \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [\frac{1}{4 x}]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{3}{3} x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.5.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{4 x}]$ .

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Step 3.1

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

$x^{2} + \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$x^{2} + \frac{1}{4} \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

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

Step 3.4

Combine $\frac{1}{4}$ and $x^{- 2}$ .

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

Step 3.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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