## Derivative of sec(x) Lesson

### Sec(x) Derivative Rule

Secant is the reciprocal of the cosine. The secant of an angle designated by a variable *x* is notated as sec(x).

The derivative rule for sec(x) is given as:

^{d}⁄_{dx}sec(x) = tan(x)sec(x)

This derivative rule gives us the ability to quickly and directly differentiate sec(x).

**Note: x may be substituted for any other variable.**

For example, the derivative ^{d}⁄_{dy}sec(y) = tan(y)sec(y), and the derivative ^{d}⁄_{dz}sec(z) = tan(z)sec(z).

INTRODUCING

#### Proof of the Derivative Rule

The sec(x) derivative rule originates from the relation that sec(x) = ^{1}⁄_{cos(x)}. Now, the first step of finding the derivative of ^{1}⁄_{cos(x)} is using the quotient rule.

- Using the quotient rule on
^{1}⁄_{cos(x)}gives us: - (
^{sin(x)}⁄_{cos(x)})(^{1}⁄_{cos(x)}) ^{sin(x)}⁄_{cos(x)}= tan(x), and^{1}⁄_{cos(x)}= sec(x)- Therefore, it simplifies to tan(x)sec(x), resulting in:
^{d}⁄_{dx}sec(x) = tan(x)sec(x)

#### Derivative Rules of the other Trigonometry Functions

Here's the derivative rules for the other five major trig functions:

^{d}⁄_{dx}sin(x) = cos(x)^{d}⁄_{dx}cos(x) = -sin(x)^{d}⁄_{dx}tan(x) = sec^{2}(x)^{d}⁄_{dx}cot(x) = -csc^{2}(x)^{d}⁄_{dx}csc(x) = -cot(x)csc(x)