Great news! We will be upgrading our calculator and lesson pages over the next few months. If you notice any issues, you can submit a contact form by clicking here.

Derivative of sec(x)

Learn about the derivative of sec(x).

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:

ddxsec(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 ddysec(y) = tan(y)sec(y), and the derivative ddzsec(z) = tan(z)sec(z).

Want unlimited access to Voovers calculators and lessons?
Join Now
100% risk free. Cancel anytime.

Proof of the Derivative Rule

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

  1. Using the quotient rule on 1cos(x) gives us:
  2. (sin(x)cos(x))(1cos(x))
  3. sin(x)cos(x) = tan(x), and 1cos(x) = sec(x)
  4. Therefore, it simplifies to tan(x)sec(x), resulting in:
    ddxsec(x) = tan(x)sec(x)

Derivative Rules of the other Trigonometry Functions

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

  • ddxsin(x) = cos(x)
  • ddxcos(x) = -sin(x)
  • ddxtan(x) = sec2(x)
  • ddxcot(x) = -csc2(x)
  • ddxcsc(x) = -cot(x)csc(x)
Learning math has never been easier.
Get unlimited access to more than 167 personalized lessons and 71 interactive calculators.
Join Voovers+ Today
100% risk free. Cancel anytime.
Scroll to Top