1)

[tex]f(x) = \frac{ {x}^{2} + 1}{x} \\ f'(x) = \frac{( {x}^{2} + 1) 'x - x'( {x}^{2} + 1) }{ {x}^{2} } = \\ \frac{2x \times x - ( {x}^{2} + 1)}{ {x}^{2} } = \\ \frac{2 {x}^{2} - {x}^{2} - 1 }{ {x}^{2} } = \frac{ {x}^{2} - 1}{ {x}^{2} } [/tex]

2)

[tex]y = - 2x \sin(x) \\ y' = ( - 2x)' \sin(x) - 2x( \sin(x) )' = \\ - 2 \sin(x) - 2x \cos(x) [/tex]

3)

[tex]y = \frac{2x - 3x {}^{2} }{3x - 4} \\ y' = \frac{(2x - 3 {x}^{2} )'(3x - 4) - (3x - 4)'(2x - 3 {x}^{2} )}{(3x - 4) {}^{2} } = \\ \frac{(2 - 6x)(3x - 4) - 3(2x - 3 {x}^{2}) }{(3x -4 ) {}^{2} } = \\ \frac{6x - 8 - 18 {x}^{2} + 24x - 6x + 9 {x}^{2} }{(3x - 4) {}^{2} } = \\ \frac{ - 9 {x}^{2} + 24x - 8}{(3x - 4) {}^{2} } [/tex]

4)

[tex]f(x) = \cos(x {}^{3} - 3 {x}^{2} ) \\ f'(x) = (3 {x}^{3 - 1} - 3 \times 2 {x}^{2 - 1} ) \times (- \sin( {x}^{3} - 3 {x}^{2} ) ) = \\ - (3 {x}^{2} - 6x) \sin( {x}^{3} - 3 {x}^{2} ) [/tex]