Ableitung/Tangens und Arkustangens/Aufgabe/Lösung

a)

${}{\begin{aligned}(\tan x)^{\prime }&={\left({\frac {\sin x}{\cos x}}\right)}^{\prime }\\&={\frac {(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}.\end{aligned}}$

b)

${}{\begin{aligned}(\arctan x)^{\prime }&={\frac {1}{\frac {1}{\cos ^{2}(\arctan x)}}}\\&={\frac {1}{\frac {\cos ^{2}(\arctan x)+\sin ^{2}(\arctan x)}{\cos ^{2}(\arctan x)}}}\\&={\frac {1}{1+\tan ^{2}(\arctan x)}}\\&={\frac {1}{1+x^{2}}}.\end{aligned}}$

Zur gelösten Aufgabe