# How do you differentiate [(2x^3) - (4x^2) + 3] / x^2 ?

Sep 24, 2015

$\frac{d}{\mathrm{dx}} \left(\frac{2 {x}^{3} - 4 {x}^{2} + 3}{x} ^ 2\right) = 2 - \frac{6}{x} ^ 3$

#### Explanation:

While it would be possible to use the quotient rule on this rational function, as it can easily be achieved in this particular case, it will be more convenient (and simpler, less prone to error, and quicker in an exam) to divide each term in the numerator by the term in the denominator and then use the sum rule.

Thus

$\frac{2 {x}^{3} - 4 {x}^{2} + 3}{x} ^ 2$

$= 2 x - 4 + 3 {x}^{- 2}$

which may be differentiated term by term using the rules for simple polynomial differentiation.

That is,

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

$= 2 - 6 {x}^{- 3}$

$= 2 - \frac{6}{x} ^ 3$