Question

1. $f(x)=\frac{x^{3}+2x}{x^{2}}$ $f(x)=\frac{3x^{2}+2}{2x}$ or $f(x)=\frac{x^{2}-2}{x^{2}}$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $f(x)=\frac{u(x)}{v(x)}$, then $f^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^2}$. Here, $u(x)=x^3 + 2x$ and $v(x)=x^2$. First, find $u^{\prime}(x)$ and $v^{\prime}(x)$. $u^{\prime}(x)=3x^2 + 2$ and $v^{\prime}(x)=2x$.

Step2: Substitute into quotient - rule formula

$f^{\prime}(x)=\frac{(3x^2 + 2)x^2-(x^3 + 2x)\times2x}{(x^2)^2}=\frac{3x^4+2x^2-(2x^4 + 4x^2)}{x^4}=\frac{3x^4+2x^2 - 2x^4-4x^2}{x^4}=\frac{x^4 - 2x^2}{x^4}=1-\frac{2}{x^2}=\frac{x^2 - 2}{x^2}$

Answer:

$f^{\prime}(x)=\frac{x^2 - 2}{x^2}$