\(f_1(x)= \frac{x + 1}{1 + x^2} \)
Ableitung mit der Quotientenregel: \( [\frac{Z}{N}]'=\frac{Z'N-ZN'}{N^2} \)
\(f'(x)= \frac{1\cdot (1 + x^2)-(x+1)\cdot 2x }{(1 + x^2)^2}=\frac{1 + x^2-2x^2-2x }{(1 + x^2)^2}=\frac{1-x^2-2x}{(1 + x^2)^2} \)
\(f_2(x) = \tan(x)\) \(x ∈ (−\frac{π}{2},\frac{π}{2})\)
\(f_2(x) =\frac{\sin(x)}{\cos(x)}\)
\(f_2(x) =\frac{\cos(x)\cdot \cos(x)-\sin(x)\cdot (-\sin(x))}{\cos^2(x)}\\=\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}=\frac{1}{\cos^2(x)}\)
\(\sin^2(x)+\cos^2(x)=1\) ist der trigonometrische Pythagoras.
