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).

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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)
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