Calculus Examples

$y = \frac{x^{3}}{12} + \frac{1}{x}$

Step 1

By the Sum Rule, the derivative of $\frac{x^{3}}{12} + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{12}] + \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{d}{d x} [\frac{x^{3}}{12}] + \frac{d}{d x} [\frac{1}{x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{x^{3}}{12}]$ .

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Step 2.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{12}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{12} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

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

$\frac{1}{12} (3 x^{2}) + \frac{d}{d x} [\frac{1}{x}]$

Step 2.3

Combine $3$ and $\frac{1}{12}$ .

$\frac{3}{12} x^{2} + \frac{d}{d x} [\frac{1}{x}]$

Step 2.4

Combine $\frac{3}{12}$ and $x^{2}$ .

$\frac{3 x^{2}}{12} + \frac{d}{d x} [\frac{1}{x}]$

Step 2.5

Cancel the common factor of $3$ and $12$ .

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Step 2.5.1

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

$\frac{3 (x^{2})}{12} + \frac{d}{d x} [\frac{1}{x}]$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $3$ out of $12$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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