Calculus Examples

$y = \frac{\sec (x)}{x}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sec (x)$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [\sec (x)] - \sec (x) \frac{d}{d x} [x]}{x^{2}}$

Step 2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{x (\sec (x) \tan (x)) - \sec (x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$\frac{x (\sec (x) \tan (x)) - \sec (x) \cdot 1}{x^{2}}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{x (\sec (x) \tan (x)) - \sec (x)}{x^{2}}$

Step 4

Factor $\sec (x)$ out of $x \sec (x) \tan (x) - \sec (x)$ .

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Step 4.1

Factor $\sec (x)$ out of $x \sec (x) \tan (x)$ .

$\frac{\sec (x) (x \tan (x)) - \sec (x)}{x^{2}}$

Step 4.2

Factor $\sec (x)$ out of $- \sec (x)$ .

$\frac{\sec (x) (x \tan (x)) + \sec (x) \cdot - 1}{x^{2}}$

Step 4.3

Factor $\sec (x)$ out of $\sec (x) (x \tan (x)) + \sec (x) \cdot - 1$ .

$\frac{\sec (x) (x \tan (x) - 1)}{x^{2}}$