# How do you find the derivative of (2x^2 +x - 3)/x?

Jan 18, 2016

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

#### Explanation:

The quotient rule states that for a function $y = \frac{f}{g}$, then $y ' = \frac{f ' g - f g '}{g} ^ 2$.

Here, we have $f = 2 {x}^{2} + x - 3$ and $f ' = 4 x + 1$. Additionally, $g = x$ and $g ' = 1$.

Applying the quotient rule, we have the derivative equalling

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

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

Jan 18, 2016

The derivative can be written: $2 + 3 {x}^{-} 2$ or $2 + \frac{3}{x} ^ 2$ or $\frac{2 {x}^{2} + 3}{x} ^ 2$. All are equivalent.

#### Explanation:

This can clearly be differentiated using the quotient rule, but it is also possible to rewrite the expression before differentiating.

$\frac{2 {x}^{2} + x - 3}{x} = \frac{2 {x}^{2}}{x} + \frac{x}{x} - \frac{3}{x} = 2 x + 1 - 3 {x}^{-} 1$

So the derivative is: $2 + 3 {x}^{-} 2$

This result can be written in other forms:

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