# Thread: Derivative of a fraction

1. ## Derivative of a fraction

I 've been trying to find the derivative of this function and it's a little confusing.

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

I started out by using the quotient rule getting:

$\displaystyle f^l(x)= \frac{(9x^2-2x)(x^\frac{1}{2})-[(3x^3-x^2+2)(\frac{1}{2}x^\frac{-1}{2})]}{x^\frac{1}{4}}$

Then multiplying this out and combining like terms:

$\displaystyle f^l(x)= \frac{\frac{15}{2}x^\frac{5}{2}-\frac{3}{2}x^\frac{3}{2}+x^\frac{-1}{2}}{x^\frac{1}{4}}$

The final solution is $\displaystyle f^l(x)= \frac{15}{2}x^\frac{3}{2}-\frac{3}{2}x^\frac{1}{2}-x^\frac{-3}{2}$

I just cant seem to get from the last step to the solution. Can anyone help? Thanks

2. $\displaystyle f^l(x)= \frac{(9x^2-2x)(x^\frac{1}{2})-[(3x^3-x^2+2)(\frac{1}{2}x^\frac{-1}{2})]}{(x^\frac{1}{2})^2}$
$\displaystyle f^l(x)= \frac{(9x^2-2x)(x^\frac{1}{2})-[(3x^3-x^2+2)(\frac{1}{2}x^\frac{-1}{2})]}{x}$
$\displaystyle f^l(x)= \frac{\frac{15}{2}x^\frac{5}{2}-\frac{3}{2}x^\frac{3}{2}+x^\frac{-1}{2}}{x}$
$\displaystyle \\ f^l(x)= \frac{15}{2}x^\frac{3}{2}-\frac{3}{2}x^\frac{1}{2}-x^\frac{-3}{2}$