241.1) ³+1/²+2, 2) ²/³+1; 3) sin/+1, 4) ln/1−

1) ³+1/²+2

Производная: \[\left(\frac{x^3+1}{x^2+2}\right)' = \frac{(3x^2)(x^2+2) - (x^3+1)(2x)}{(x^2+2)^2} = \frac{3x^4 + 6x^2 - 2x^4 - 2x}{(x^2+2)^2} = \frac{x^4 + 6x^2 - 2x}{(x^2+2)^2}\]

2) ²/³+1

Производная: \[\left(\frac{x^2}{x^3+1}\right)' = \frac{(2x)(x^3+1) - x^2(3x^2)}{(x^3+1)^2} = \frac{2x^4 + 2x - 3x^4}{(x^3+1)^2} = \frac{-x^4 + 2x}{(x^3+1)^2}\]

3) sin/+1

Производная: \[\left(\frac{\sin x}{x+1}\right)' = \frac{(\cos x)(x+1) - \sin x}{(x+1)^2} = \frac{x \cos x + \cos x - \sin x}{(x+1)^2}\]

4) ln/1−

Производная: \[\left(\frac{\ln x}{1-x}\right)' = \frac{\frac{1}{x}(1-x) - (\ln x)(-1)}{(1-x)^2} = \frac{\frac{1}{x} - 1 + \ln x}{(1-x)^2} = \frac{1 - x + x \ln x}{x(1-x)^2}\]