Question

$$\frac{(x^3+1)(x-2)}{x^2}$$ 

Answer 1

$$\frac{d}{dx}\left(\frac{(x^3+1)(x-2)}{x^2}\right)$$
$$=\frac{d}{dx}\left(\frac{x^4-2x^3+x-2}{x^2}\right)$$
$$=\frac{d}{dx}\left(\frac{x^4}{x^2}\right)-2\frac{d}{dx}\left(\frac{x^3}{x^2}\right)+\frac{d}{dx}\left(\frac{x}{x^2}\right)-\frac{d}{dx}\left(\frac{2}{x^2}\right)$$
$$=\frac{d}{dx}\left(x^2\right)-2\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(x^{-1}\right)-2\frac{d}{dx}\left(x^{-2}\right)$$
$$=2x-2-\frac{1}{x^2}-2(-2)x^{-3}$$
$$=2x-2-\frac{1}{x^2}+\frac{4}{x^3}$$