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⁄_dxsec(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⁄_dysec(y) = tan(y)sec(y), and the derivative ^d⁄_dzsec(z) = tan(z)sec(z).

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.

1. Using the quotient rule on ¹⁄_cos(x) gives us:
2. (^sin(x)⁄_cos(x))(¹⁄_cos(x))
3. ^sin(x)⁄_cos(x) = tan(x), and ¹⁄_cos(x) = sec(x)
4. Therefore, it simplifies to tan(x)sec(x), resulting in:
   ^d⁄_dxsec(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⁄_dxsin(x) = cos(x)
• ^d⁄_dxcos(x) = -sin(x)
• ^d⁄_dxtan(x) = sec²(x)
• ^d⁄_dxcot(x) = -csc²(x)
• ^d⁄_dxcsc(x) = -cot(x)csc(x)