Calculus

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Tutors/teachers: Nikkolas
Tutor and Aerospace Engineer

# 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:

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

Result :

Worksheet 1