\(y(x) = \tan(x) = \frac{\sin(x)}{\cos(x)} = \sin(x) \cdot (\cos(x))^{-1}\)

\(y'(x) = \cos(x) \cdot (\cos(x))^{-1} + \sin(x) \cdot (\cos(x))^{-2} \sin(x) = 1 + (\sin(x))^2 \cdot (\cos(x))^{-2} = \)

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

\(y=\sqrt{3x}-\sqrt[3]{x^2}\)

setze

\(g(x)= 3x, \ h(x)= x^{\frac 12}, \ k(x)= x^{\frac 23}\)

\(g'(x)=3 \\ h'(x)=\frac 13 x^{-\frac 12} \\ k'(x)= \frac 23 x ^{-\frac 13}\)

\(y'= h'(g(x))g'(x) - k'(x)\)

\(y'= \frac 12 (3x)^{-\frac 12} \cdot 3 - \frac 23 x^{-\frac 13} =\\ \frac {\sqrt 3}{2\sqrt{x}}-\frac{2}{3\sqrt[3]x}\)

\(y=\sqrt{\sin x}\)

setze \(f(x)= x^{1/2}\) , \(g(x)= \sin x\)

\(f'(x)=\frac 12 x^{-1/2}\) , \(g'(x)=\cos x\)

\(y'= \frac 12 (\sin x)^{-1/2} \cos x = \frac{\cos x}{2\sqrt{\sin x}}\)

\(y=\sqrt[4]{3x^3}-\sqrt[3]{2x}+\sqrt{2x^3}\)

1.   \(\sqrt[4]{3x^3}\)  

setze \(f(x)=x^{1/4}\) \(g(x)=3x^3\)

\(f'(x)=\frac{1}{4} x^{-3/4}\) , \(g'(x)=9x^2\)

\(\sqrt[4]{3x^3}'=\frac 14 (3x^3)^{-3/4} \cdot 9x^2 =\\ \frac 94 3^{-3/4}x^{-1/4} = \frac{9}{4 \cdot \sqrt[4] {27x}}\)

2. \(\sqrt[3]{2x}\)

setze \(f(x)=x^{1/3}\) \(g(x)=2x\)

\(f'(x)=\frac 13 x^{-2/3}\), \(g'(x)=2\)

\(\sqrt[3]{2x}'= \frac 13 (2x)^{-2/3} \cdot 2 = \frac{\sqrt[3]2}{3 \cdot \sqrt[3]{x^2}}\)

3. \(\sqrt{2x^3}\)

setze \(f(x)= x^{1/2}\) , \(g(x)=2x^3\)

\(f'(x)= \frac 12 x^{-1/2}\)  , \(g'(x)=6x^2\)

\(\sqrt{2x^3}'= \frac 12 (2x^3)^{-1/2} \cdot 6x^2 = \frac {3 \sqrt x}{\sqrt 2}\)

\(y'=\frac{9}{4\cdot\sqrt[4]{27x}}-\frac{\sqrt[3]{2}}{3\cdot\sqrt[3]{x^2}}+\frac{3\cdot\sqrt{x}}{\sqrt{2}}\)

\(y=\left(\frac{2x}{3x-1}\right)^3\)

setze \(f(x)= x^3\),   \(g(x)= \frac{2x}{3x-1}\)

\(f'(x)= 3x^2\)

\(g'(x)=\frac{2 \cdot (3x-1)-2x \cdot 3}{(3x-1)^2} = \frac{-2}{(3x-1)^2}\)

\(y'= 3\left(\frac {2x}{3x-1}\right)^2 \cdot \frac{-2}{(3x-1)^2} = \frac{-24x^2}{(3x-1)^4}\)

\(y(x) = \frac{-3+x}{5x+2} \\ y(x) = (-3+x) \cdot \frac{1}{5x+2} \\ y'(x) = 1 \cdot \frac{1}{5x+2} + (-3+x) \cdot (-1) \cdot \frac{1}{(5x+2)^2} \cdot 5 \\\)

\( y'(x) = \frac{1}{5x+2} + \frac{(3-x)\cdot 5}{(5x+2)^2} \\ y'(x) = \frac{5x + 2 + 15 - 5x}{(5x+2)^2} \\ y'(x) = \frac{17}{(5x+2)^2} \)

\(y=(2x-5)^3\)

setze

\(g(x)=2x-5,\ h(x)=x^3\)

\(g'(x)=2\)

\(h'(x)=3x^2\)

\(y'=h'(g(x))g'(x)=3(2x-5)^2 \cdot 2 = 6(2x-5)^2\)

\(e^{y-x^2} = \frac{1}{x} \\ y - x^2 = \ln(\frac{1}{x}) \\ y (x) = x^2 + \ln(\frac{1}{x})\)

\(y' (x) = 2x + \frac{1}{\frac{1}{x}} \cdot (-\frac{1}{x^2}) \\ y'(x) = 2x - \frac{1}{x}\)