Derivative of Tangent Function

Theorem

$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$

when $\cos x \ne 0$.

Corollary 1

$\map {\dfrac \d {\d x} } {\tan a x} = a \sec^2 a x$

Corollary 2

$\dfrac \d {\d x} \tan x = 1 + \tan^2 x$

Corollary 3

$\map {\dfrac \d {\d x} } {\tan a x} = a \paren {\tan^2 a x + 1}$

Proof 1

From the definition of the tangent function:

$\tan x = \dfrac {\sin x} {\cos x}$

From Derivative of Sine Function:

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

From Derivative of Cosine Function:

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

Then:

\(\ds \map {\dfrac \d {\d x} } {\tan x}\)

\(=\)

\(\ds \frac {\cos x \cos x - \sin x \paren {-\sin x} } {\cos^2 x}\)

Quotient Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \frac {\cos^2 x + \sin^2 x} {\cos^2 x}\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos^2 x}\)

Sum of Squares of Sine and Cosine

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

The result follows from the Secant is Reciprocal of Cosine.

$\blacksquare$

Proof 2

\(\ds \map {\frac \d {\d x} } {\tan x}\)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\map \tan {x + h} - \tan x} h\)

Definition of Derivative of Real Function at Point

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h} {1 - \tan x \tan h} - \tan x} h\)

Tangent of Sum

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h - \tan x + \tan^2 x \tan h} {1 - \tan x \tan h} } h\)

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\tan h + \tan^2 x \tan h} {h \paren {1 - \tan x \tan h} }\)

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {1 + \tan^2 x} {1 - \tan x \tan h} \cdot \lim_{h \mathop \to 0} \frac {\tan h} h\)

Product Rule for Limits of Real Functions

\(\ds \)

\(=\)

\(\ds \frac {1 + \tan^2 x} {1 - \tan x \tan 0} \cdot 1\)

Limit of $\dfrac {\tan x} x$ at Zero

\(\ds \)

\(=\)

\(\ds 1 + \tan^2 x\)

Tangent of Zero

\(\ds \)

\(=\)

\(\ds \sec^2 x\)

Difference of Squares of Secant and Tangent

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos^2 x}\)

Secant is Reciprocal of Cosine ($\cos x \ne 0$)

$\blacksquare$

Also see

• Derivative of Sine Function
• Derivative of Cosine Function

• Derivative of Cotangent Function

• Derivative of Secant Function
• Derivative of Cosecant Function

Sources

• 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $6$.
• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $3.$ Trigonometric functions
• 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