Derivative of Tangent Function

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Theorem

$D_x \left({\tan x}\right) = \sec^2 x = \dfrac 1 {\cos^2 x}$

when $\cos x \ne 0$.


Corollary

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


Proof 1

From the definition of the tangent function:

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

From Derivative of Sine Function:

$D_x \left({\sin x}\right) = \cos x$

From Derivative of Cosine Function:

$D_x \left({\cos x}\right) = -\sin x$


Then:

\(\displaystyle D_x \left({\tan x}\right)\) \(=\) \(\displaystyle \frac {\cos x \cos x - \sin x \left({-\sin x}\right)} {\cos^2 x}\) Quotient Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cos^2 x + \sin^2 x} {\cos^2 x}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\cos^2 x}\) 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

\(\displaystyle \map {D_x} {\tan x}\) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map \tan {x + h} - \tan x} h\) Definition of Derivative of Real Function at Point
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h} {1 - \tan x \tan h} - \tan x} h\) Tangent of Sum
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h - \tan x + \tan^2 x \tan h} {1 - \tan x \tan h} } h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\tan h + \tan^2 x \tan h} {h \paren {1 - \tan x \tan h} }\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {1 + \tan^2 x} {1 - \tan x \tan h} \cdot \lim_{h \mathop \to 0} \frac {\tan h} h\) Product Rule for Limits of Functions
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + \tan^2 x} {1 - \tan x \tan 0} \cdot 1\) Limit of Tan X over X
\(\displaystyle \) \(=\) \(\displaystyle 1 + \tan^2 x\) Tangent of Zero
\(\displaystyle \) \(=\) \(\displaystyle \sec^2 x\) Difference of Squares of Secant and Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\cos^2 x}\) Secant is Reciprocal of Cosine ($\cos x \ne 0$)


$\blacksquare$


Sources