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:

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

From Derivative of Cosine Function:

$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$

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 $\cos x \ne 0$.

$\blacksquare$

Also see

Derivative of Cosecant Function

Sources

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives