# What is the derivative of  f(x) = (x^3-3)/x?

Mar 15, 2018

The derivative of $f \left(x\right)$ is $\frac{2 {x}^{3} + 3}{{x}^{2}}$.

#### Explanation:

Using the quotient rule of derivatives:

$\frac{d}{\mathrm{dx}} \left(\frac{f \left(x\right)}{g \left(x\right)}\right) = \frac{\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right) \cdot g \left(x\right) - f \left(x\right) \cdot \frac{d}{\mathrm{dx}} \left(g \left(x\right)\right)}{{\left(g \left(x\right)\right)}^{2}}$

Here's our expression:

$\textcolor{w h i t e}{=} \frac{d}{\mathrm{dx}} \left(\frac{{x}^{3} - 3}{x}\right)$

$= \frac{\frac{d}{\mathrm{dx}} \left({x}^{3} - 3\right) \cdot x - \left({x}^{3} - 3\right) \cdot \frac{d}{\mathrm{dx}} \left(x\right)}{{x}^{2}}$

$= \frac{\left(3 {x}^{2} - 0\right) \cdot x - \left({x}^{3} - 3\right) \cdot 1}{{x}^{2}}$

$= \frac{3 {x}^{3} - {x}^{3} + 3}{{x}^{2}}$

$= \frac{2 {x}^{3} + 3}{{x}^{2}}$

This is the derivative. Hope this helped!