The first derivative of \( \tan (x)\)
is calculated using the derivative of \( \sin x \) and \( \cos x \) and the quotient rule of derivatives.
Calculation of the Derivative of tan x
A trigonometric identity relating \( \tan x \), \( \sin x \) and \( \cos x \) is given by
\[ \tan x = \dfrac { \sin x }{ \cos x } \]
One way to find the derivative of \( \tan x \) is to use the quotient rule of differentiation; hence
\( \displaystyle {\dfrac {d}{dx} \tan x = \dfrac {d}{dx} (\dfrac{\sin x }{\cos x}) = \dfrac{{ (\dfrac {d}{dx}\sin x) }{ \cos x } - \sin x (\dfrac {d}{dx} \cos x) }{\cos^2 x}} \)
Use the formulae for the derivative of the trigonometric functions \( \sin x \) and \( \cos x \) given by \( \dfrac {d}{dx}\sin x = \cos x \) and \( \dfrac {d}{dx}\cos x = - \sin x \) and substitute to obtain
\( \displaystyle {\dfrac {d}{dx} \tan x = (\dfrac{{ \cos x \cos x } - \sin x (-\sin x) }{\cos^2 x}} \)
Simplify
\( \displaystyle {= \dfrac{ \cos^2 x + \sin^2 x } {\cos^2 x} = \dfrac{ 1 }{\cos^2 x} = \sec^2 x }\)
conclusion
\[ \displaystyle {\dfrac {d}{dx} \tan x = \sec^2 x} \]
Graph of tan x and its Derivative
The graphs of \( \tan(x) \) and its derivative are shown below.
Derivative of the Composite Function tan (u(x))
We now have a composite function which is a function (tan) of another function (u). Use the chain rule of differentiation to write