# Derivative of Secant Function

## Theorem

$\map {\dfrac \d {\d x} } {\sec x} = \sec x \tan x$

where $\cos x \ne 0$.

## Proof 1

From the definition of the secant function:

$\sec x = \dfrac 1 {\cos x} = \paren {\cos x}^{-1}$
$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$

Then:

 $\ds \map {\dfrac \d {\d x} } {\sec x}$ $=$ $\ds \map {\dfrac \d {\d x} } {\paren {\cos x}^{-1} }$ Exponent Laws $\ds$ $=$ $\ds \paren {-\sin x} \paren {-\cos^{-2} x}$ Chain Rule for Derivatives, Power Rule $\ds$ $=$ $\ds \frac 1 {\cos x} \frac {\sin x} {\cos x}$ Exponent Laws $\ds$ $=$ $\ds \sec x \tan x$ Definition of Real Secant Function and Definition of Real Tangent Function

This is valid only when $\cos x \ne 0$.

$\blacksquare$

## Proof 2

 $\ds \map {\dfrac \d {\d x} } {\sec x}$ $=$ $\ds \map {\dfrac \d {\d x} } {\dfrac 1 {\cos x} }$ Definition of Real Secant Function $\ds$ $=$ $\ds \dfrac {\cos x \map {\frac \d {\d x} } 1 - 1 \map {\frac \d {\d x} } {\cos x} } {\cos^2 x}$ Quotient Rule for Derivatives $\ds$ $=$ $\ds \dfrac {0 - \paren {-\sin x} } {\cos^2 x}$ Derivative of Cosine Function, Derivative of Constant $\ds$ $=$ $\ds \sec x \tan x$ Definition of Real Secant Function, Definition of Real Tangent Function

This is valid only when $\cos x \ne 0$.

$\blacksquare$