Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$- 3 x^{2} + 1 - x^{- 2}$

Step 5

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

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

Step 6

Reorder terms.

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