Calculus Examples

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

Step 1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.1

Factor $x$ out of $2 x$ .

$\frac{d}{d x} [\frac{x \cdot 2}{x^{2}}]$

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

$\frac{d}{d x} [\frac{x \cdot 2}{x \cdot x}]$

Step 1.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x \cdot 2}{x \cdot x}]$

Step 1.2.3

Rewrite the expression.

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

Step 2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 3

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

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

Step 4

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

$2 (- x^{- 2})$

Step 5

Multiply $- 1$ by $2$ .

$- 2 x^{- 2}$

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Combine terms.

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Step 6.2.1

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

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

Step 6.2.2

Move the negative in front of the fraction.

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