Derivative of Tangent Function/Corollary 3

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Corollary to Derivative of Tangent Function

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


Proof

\(\ds \map {\dfrac \d {\d x} } {\tan x}\) \(=\) \(\ds \sec^2 x\) Derivative of $\tan x$
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\tan a x}\) \(=\) \(\ds a \sec^2 a x\) Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds a \paren {\tan^2 a x + 1}\) Difference of Squares of Secant and Tangent

$\blacksquare$


Also see