Calculus

##### Related Content

# Derivative of sec(x)

## 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).*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).

### Proof of the Derivative Rule

The sec(x) derivative rule is 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)