Derivative of tan x
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 xA 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\( \displaystyle \dfrac{d}{dx} \tan (u(x)) = (\dfrac{d}{du} \tan u) (\dfrac{d}{dx} u ) \) Simplify \( = \sec^2 u \dfrac{d}{dx} u \) Conclusion \[ \displaystyle \dfrac{d}{dx} \tan (u(x)) = \sec^2 u \dfrac{d}{dx} u \]
Example 1
Solution to Example 1
More References and linksRules of Differentiation of Functions in Calculus.Trigonometric Identities and Formulas. Derivatives of the Trigonometric Functions. Chain Rule of Differentiation in Calculus. |