# Derivative of Secant Function

## Theorem

$D_x \left({\sec x}\right) = \sec x \tan x$

where $\cos x \ne 0$.

## Proof

From the definition of the secant function:

$\sec x = \dfrac 1 {\cos x} = \left({\cos x}\right)^{-1}$
$D_x \left({\cos x}\right) = -\sin x$

Then:

 $\displaystyle D_x \left({\sec x}\right)$ $=$ $\displaystyle D_x \left({ \left({\cos x}\right)^{-1} }\right)$ Exponent Laws $\displaystyle$ $=$ $\displaystyle \left({-\sin x}\right) \left({-\cos^{-2} x}\right)$ Chain Rule, Power Rule $\displaystyle$ $=$ $\displaystyle \frac 1 {\cos x} \frac {\sin x} {\cos x}$ Exponent Laws $\displaystyle$ $=$ $\displaystyle \sec x \tan x$ Definitions of secant and tangent

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

$\blacksquare$