Given:
The given function is:
[tex]\dfrac{(x^3+1)(x-2)}{x^2}[/tex]
To find:
The differentiation of the given function.
Solution:
Consider the given function,
[tex]y=\dfrac{(x^3+1)(x-2)}{x^2}[/tex]
It can be written as:
[tex]y=\dfrac{(x^3)(x)+(x^3)(-2)+(1)(x)+(1)(-2)}{x^2}[/tex]
[tex]y=\dfrac{x^4-2x^3+x-2}{x^2}[/tex]
[tex]y=\dfrac{x^4}{x^2}-\dfrac{2x^3}{x^2}+\dfrac{x}{x^2}-\dfrac{2}{x^2}[/tex]
[tex]y=x^2-2x+\dfrac{1}{x}-\dfrac{2}{x^2}[/tex]
Differentiate with respect to x.
[tex]y'=\dfrac{d}{dx}(x^2)-\dfrac{d}{dx}(2x)+\dfrac{d}{dx}(x^{-1})-\dfrac{d}{dx}\2(x^{-2})[/tex]
[tex]y'=2x-2(1)+(-x^{-2})-2(-2x^{-3})[/tex]
[tex]y'=2x-2-\dfrac{1}{x^2}+\dfrac{4}{x^3}[/tex]
Therefore, the differentiation of the given function is [tex]2x-2-\dfrac{1}{x^2}+\dfrac{4}{x^3}[/tex].
