Calculus Examples

$f (x) = 2 x - \frac{1}{x}$

Step 1

By the Sum Rule, the derivative of $2 x - \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{x}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 - - x^{- 2} + \frac{1}{x} \cdot 0$

Step 3.5

Multiply $- 1$ by $- 1$ .

$2 + 1 x^{- 2} + \frac{1}{x} \cdot 0$

Step 3.6

Multiply $x^{- 2}$ by $1$ .

$2 + x^{- 2} + \frac{1}{x} \cdot 0$

Step 3.7

Multiply $\frac{1}{x}$ by $0$ .

$2 + x^{- 2} + 0$

Step 3.8

Add $x^{- 2}$ and $0$ .

$2 + x^{- 2}$

Step 4

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

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

Step 5

Reorder terms.

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