Calculus Examples

$\frac{5 x}{2 x^{2}}$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

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

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Step 2.1

Factor $x$ out of $5 x$ .

$\frac{d}{d x} [\frac{x \cdot 5}{2 x^{2}}]$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$\frac{d}{d x} [\frac{x \cdot 5}{x (2 x)}]$

Step 2.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x \cdot 5}{x (2 x)}]$

Step 2.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{5}{2 x}]$

Step 3

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5}{2 x}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{5}{2} \frac{d}{d x} [\frac{1}{x}]$

Step 4

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

$\frac{5}{2} \frac{d}{d x} [x^{- 1}]$

Step 5

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

$\frac{5}{2} (- x^{- 2})$

Step 6

Combine fractions.

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Step 6.1

Combine $\frac{5}{2}$ and $x^{- 2}$ .

$- \frac{5 x^{- 2}}{2}$

Step 6.2

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

$- \frac{5}{2 x^{2}}$