aufgabe 2c

$$f_3(x)=\frac{\sin \left(\sqrt{x}\right)}{1+\cos \left(\sqrt{x}\right)}$$

Ableitung §

$$\begin{array}{rl}\frac{d}{dx}{f}_3& =\frac{\frac{d}{dx}\left(\sin \left(\sqrt{x}\right)\right)\cdot \left(1+\cos \left(\sqrt{x}\right)\right)-\left(\sin \left(\sqrt{x}\right)\right)\cdot \frac{d}{dx}\left(1+\cos \left(\sqrt{x}\right)\right)}{{\left(1+\cos \left(\sqrt{x}\right)\right)}^2}\\ & =\frac{\left(\cos \left(\sqrt{x}\right)\cdot {\left(\sqrt{x}\right)}^{\prime }\right)\cdot \left(1+\cos \left(\sqrt{x}\right)\right)-\left(\sin \left(\sqrt{x}\right)\right)\cdot \left(-\sin \left(\sqrt{x}\right)\cdot {\left(\sqrt{x}\right)}^{\prime }\right)}{{\left(1+\cos \left(\sqrt{x}\right)\right)}^2}\\ & =\frac{\left(\cos \left(\sqrt{x}\right)\cdot \frac{1}{2\sqrt{x}}\right)\cdot \left(1+\cos \left(\sqrt{x}\right)\right)-\left(\sin \left(\sqrt{x}\right)\right)\cdot \left(-\sin \left(\sqrt{x}\right)\cdot \frac{1}{2\sqrt{x}}\right)}{{\left(1+\cos \left(\sqrt{x}\right)\right)}^2}\\ & =\frac{\frac{\left(\cos (\sqrt{x})\right)\left(1+\cos (\sqrt{x})\right)}{2\sqrt{x}}+\frac{{\sin }^2(\sqrt{x})}{2\sqrt{x}}}{(1+\cos (\sqrt{x}){)}^2}\\ & =\frac{\cos (\sqrt{x})+\stackrel{=1}{\overbrace{{\cos }^2(\sqrt{x})+{\sin }^2(\sqrt{x})}}}{2\sqrt{x}\cdot (1+\cos (\sqrt{x}){)}^2}\\ & =\frac{\cos (\sqrt{x})+1}{2\sqrt{x}\cdot (1+\cos (\sqrt{x}){)}^2}\\ & =\frac{1}{2\sqrt{x}\cdot (1+\cos (\sqrt{x}))}\end{array}$$

2.21 (vii)

$${\cos }^2(x)+{\sin }^2(x)=1$$