1) y=(2x^2-3x+1)/x = 2x-3+1/x
y'=2-1/x^2
2) S=6√(t^2+t+2)
S'=6* 1/2√(t^2+t+2) * (t^2+t+2)' = 3/√(t^2+t+2) * (2t+1) = 3(2t+1)/√(t^2+t+2)
S'(1)=3*3/√(1+1+2)=9/√4=9/2=4.5
3) y=(2x^2+1)/(x^2-1)
y'=((2x^2+1)'(x^2-1)-(x^2-1)'(2x^2+1))/(x^2-1)^2 = (4x(x^2-1) -2x(2x^2+1))/(x^2-1)^2 = (4x^3-4x - 4x^3 - 2x)/(x^2-1)^2 = -6x/(x^2-1)^2
4) f(x)= 1/(6x-1)^5
f'(x) = -5/(6x-1)^6 * (6x-1)' = -30/(6x-1)^6
5) f(x)=(x+1)√(x+1)=√(x+1)^3
f'(x) = 1/2√(x+1)^3 * ((x+1)^3)'=1/2√(x+1)^3 * 3(x+1)^2 = 3(x+1)^2/2(x+1)√(x+1) = 3(x+1)/2√(x+1) = 3(x+1)√(x+1)/2(x+1) = 3√(x+1)/2
f'(2) = 3√3/2
6) y=(3x-5)(2-4x)=6x-12x^2-10+20x=-12x^2+26x-10
y'=-24x+26
7) y=0.3(2x^2-1)^5
y'=0.3*5(2x^2-1)^4*4x=6x(2x^2-1)^4
8) y=sin^2(x)
y'=2sin(x)*cosx=2sinxcosx=sin2x